{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r05-connected-topological-space-0001","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"40f0a47e674c6063249347a6dfdd27391193ead0e006ba84e5852b28244b9743","family_id":"qf-mod34-connected-separation-audit","seed":202611035001,"semantic_template":"qf-mod34-connected-separation-audit/{\"left\":[0,1],\"n\":5,\"right\":[2,3,4],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2],[0,1],[0]]}"},"fields":{"title":"A Separation Must Meet Every Condition on the Actual Space","stem":"Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Proposed left=[0, 1], right=[2, 3, 4]. Give (left,right,valid_separation?,classification), with classification=disconnected_separation or not_a_separation.","answer":{"kind":"expression","value":"([0, 1], [2, 3, 4], No, not_a_separation)","explanation":"Check openness,nonemptiness,disjointness and full coverage together. A valid proposal witnesses disconnection. An invalid proposal is merely not a separation; it does not establish connectedness of the whole space. The exact result, retaining the requested order, is ([0, 1], [2, 3, 4], No, not_a_separation)."},"skills":["A Separation Must Meet Every Condition on the Actual Space"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r05-connected-topological-space-0002","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"882df4a46dc53831d067c87fe0843e368f67f734a737cf9424900f265dd0a492","family_id":"qf-mod34-connected-separation-audit","seed":202611035002,"semantic_template":"qf-mod34-connected-separation-audit/{\"left\":[0,1],\"n\":5,\"right\":[2,3,4],\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,2],[0,1],[0]]}"},"fields":{"title":"A Separation Must Meet Every Condition on the Actual Space","stem":"Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Proposed left=[0, 1], right=[2, 3, 4]. Give (left,right,valid_separation?,classification), with classification=disconnected_separation or not_a_separation.","answer":{"kind":"expression","value":"([0, 1], [2, 3, 4], No, not_a_separation)","explanation":"Check openness,nonemptiness,disjointness and full coverage together. A valid proposal witnesses disconnection. An invalid proposal is merely not a separation; it does not establish connectedness of the whole space. The exact result, retaining the requested order, is ([0, 1], [2, 3, 4], No, not_a_separation)."},"skills":["A Separation Must Meet Every Condition on the Actual Space"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r05-connected-topological-space-0003","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"da3cbaa9e317525d4f2f268418e3c38714973d91c309ad26acf0f2659a7f6945","family_id":"qf-mod34-connected-separation-audit","seed":202611035003,"semantic_template":"qf-mod34-connected-separation-audit/{\"left\":[0,1],\"n\":5,\"right\":[2,3,4],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,3],[0,1],[0]]}"},"fields":{"title":"A Separation Must Meet Every Condition on the Actual Space","stem":"Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 1], [0, 1, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Proposed left=[0, 1], right=[2, 3, 4]. Give (left,right,valid_separation?,classification), with classification=disconnected_separation or not_a_separation.","answer":{"kind":"expression","value":"([0, 1], [2, 3, 4], No, not_a_separation)","explanation":"Check openness,nonemptiness,disjointness and full coverage together. A valid proposal witnesses disconnection. An invalid proposal is merely not a separation; it does not establish connectedness of the whole space. The exact result, retaining the requested order, is ([0, 1], [2, 3, 4], No, not_a_separation)."},"skills":["A Separation Must Meet Every Condition on the Actual Space"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r05-connected-topological-space-0004","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"02dc4009fb698330ac1cc24c328ef7867dfb1a14fb8004dfa3cfe00803043918","family_id":"qf-mod34-connected-separation-audit","seed":202611035004,"semantic_template":"qf-mod34-connected-separation-audit/{\"left\":[0,1],\"n\":5,\"right\":[2,3,4],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,3],[0,1],[0]]}"},"fields":{"title":"A Separation Must Meet Every Condition on the Actual Space","stem":"Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 1], [0, 1, 3], [0, 1, 3, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Proposed left=[0, 1], right=[2, 3, 4]. Give (left,right,valid_separation?,classification), with classification=disconnected_separation or not_a_separation.","answer":{"kind":"expression","value":"([0, 1], [2, 3, 4], No, not_a_separation)","explanation":"Check openness,nonemptiness,disjointness and full coverage together. A valid proposal witnesses disconnection. An invalid proposal is merely not a separation; it does not establish connectedness of the whole space. The exact result, retaining the requested order, is ([0, 1], [2, 3, 4], No, not_a_separation)."},"skills":["A Separation Must Meet Every Condition on the Actual Space"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r05-connected-topological-space-0005","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"34e4286ddad9cf9779bb90778d001f45abc1f68b76ebe4356529762aa28032cb","family_id":"qf-mod34-connected-separation-audit","seed":202611035005,"semantic_template":"qf-mod34-connected-separation-audit/{\"left\":[0,1],\"n\":5,\"right\":[2,3,4],\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,4],[0,1],[0]]}"},"fields":{"title":"A Separation Must Meet Every Condition on the Actual Space","stem":"Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 1], [0, 1, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Proposed left=[0, 1], right=[2, 3, 4]. Give (left,right,valid_separation?,classification), with classification=disconnected_separation or not_a_separation.","answer":{"kind":"expression","value":"([0, 1], [2, 3, 4], No, not_a_separation)","explanation":"Check openness,nonemptiness,disjointness and full coverage together. A valid proposal witnesses disconnection. An invalid proposal is merely not a separation; it does not establish connectedness of the whole space. The exact result, retaining the requested order, is ([0, 1], [2, 3, 4], No, not_a_separation)."},"skills":["A Separation Must Meet Every Condition on the Actual Space"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r05-connected-topological-space-0006","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4d454a8ec5018b89dd2d9fffeb86093c3bb7b0cb2108fc0dbf6bd0c1bc5ac517","family_id":"qf-mod34-connected-separation-audit","seed":202611035006,"semantic_template":"qf-mod34-connected-separation-audit/{\"left\":[0,1],\"n\":5,\"right\":[2,3,4],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,4],[0,1],[0]]}"},"fields":{"title":"A Separation Must Meet Every Condition on the Actual Space","stem":"Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 1], [0, 1, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Proposed left=[0, 1], right=[2, 3, 4]. Give (left,right,valid_separation?,classification), with classification=disconnected_separation or not_a_separation.","answer":{"kind":"expression","value":"([0, 1], [2, 3, 4], No, not_a_separation)","explanation":"Check openness,nonemptiness,disjointness and full coverage together. A valid proposal witnesses disconnection. An invalid proposal is merely not a separation; it does not establish connectedness of the whole space. The exact result, retaining the requested order, is ([0, 1], [2, 3, 4], No, not_a_separation)."},"skills":["A Separation Must Meet Every Condition on the Actual Space"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r05-connected-topological-space-0007","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7895d24c7a60412cbe51496a5620e9f59aad5bfef084282148b3fbf51e2a5584","family_id":"qf-mod34-connected-separation-audit","seed":202611035007,"semantic_template":"qf-mod34-connected-separation-audit/{\"left\":[0,2],\"n\":5,\"right\":[1,3,4],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2],[0,2],[0]]}"},"fields":{"title":"A Separation Must Meet Every Condition on the Actual Space","stem":"Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 2], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Proposed left=[0, 2], right=[1, 3, 4]. Give (left,right,valid_separation?,classification), with classification=disconnected_separation or not_a_separation.","answer":{"kind":"expression","value":"([0, 2], [1, 3, 4], No, not_a_separation)","explanation":"Check openness,nonemptiness,disjointness and full coverage together. A valid proposal witnesses disconnection. An invalid proposal is merely not a separation; it does not establish connectedness of the whole space. The exact result, retaining the requested order, is ([0, 2], [1, 3, 4], No, not_a_separation)."},"skills":["A Separation Must Meet Every Condition on the Actual Space"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r05-connected-topological-space-0008","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"844090bd4467c2cb58ba057ef732248978f87b7f40f893b02c251934eb52337f","family_id":"qf-mod34-connected-separation-audit","seed":202611035008,"semantic_template":"qf-mod34-connected-separation-audit/{\"left\":[0,2],\"n\":5,\"right\":[1,3,4],\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,2],[0,2],[0]]}"},"fields":{"title":"A Separation Must Meet Every Condition on the Actual Space","stem":"Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 2], [0, 1, 2], [0, 1, 2, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Proposed left=[0, 2], right=[1, 3, 4]. Give (left,right,valid_separation?,classification), with classification=disconnected_separation or not_a_separation.","answer":{"kind":"expression","value":"([0, 2], [1, 3, 4], No, not_a_separation)","explanation":"Check openness,nonemptiness,disjointness and full coverage together. A valid proposal witnesses disconnection. An invalid proposal is merely not a separation; it does not establish connectedness of the whole space. The exact result, retaining the requested order, is ([0, 2], [1, 3, 4], No, not_a_separation)."},"skills":["A Separation Must Meet Every Condition on the Actual Space"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r05-connected-topological-space-0009","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c11449b24a207f5f5b3871522189157542d04d61a76abdfb2e2ad2e0e9fc61b9","family_id":"qf-mod34-connected-separation-audit","seed":202611035009,"semantic_template":"qf-mod34-connected-separation-audit/{\"left\":[0,2],\"n\":5,\"right\":[1,3,4],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,2,3],[0,2],[0]]}"},"fields":{"title":"A Separation Must Meet Every Condition on the Actual Space","stem":"Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 2], [0, 2, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Proposed left=[0, 2], right=[1, 3, 4]. Give (left,right,valid_separation?,classification), with classification=disconnected_separation or not_a_separation.","answer":{"kind":"expression","value":"([0, 2], [1, 3, 4], No, not_a_separation)","explanation":"Check openness,nonemptiness,disjointness and full coverage together. A valid proposal witnesses disconnection. An invalid proposal is merely not a separation; it does not establish connectedness of the whole space. The exact result, retaining the requested order, is ([0, 2], [1, 3, 4], No, not_a_separation)."},"skills":["A Separation Must Meet Every Condition on the Actual Space"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r05-connected-topological-space-0010","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"831df5ea8b3074978626b99565974707611d10954ae1c36f6ca1f1e5da5e0e99","family_id":"qf-mod34-connected-separation-audit","seed":202611035010,"semantic_template":"qf-mod34-connected-separation-audit/{\"left\":[0,2],\"n\":5,\"right\":[1,3,4],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,3],[0,2],[0]]}"},"fields":{"title":"A Separation Must Meet Every Condition on the Actual Space","stem":"Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Proposed left=[0, 2], right=[1, 3, 4]. Give (left,right,valid_separation?,classification), with classification=disconnected_separation or not_a_separation.","answer":{"kind":"expression","value":"([0, 2], [1, 3, 4], No, not_a_separation)","explanation":"Check openness,nonemptiness,disjointness and full coverage together. A valid proposal witnesses disconnection. An invalid proposal is merely not a separation; it does not establish connectedness of the whole space. The exact result, retaining the requested order, is ([0, 2], [1, 3, 4], No, not_a_separation)."},"skills":["A Separation Must Meet Every Condition on the Actual Space"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r05-connected-topological-space-0011","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f7308831f78d0dc2833ba017e4e774ea0b0fc54685e632a893d4033bc18194ff","family_id":"qf-mod34-connected-separation-audit","seed":202611035011,"semantic_template":"qf-mod34-connected-separation-audit/{\"left\":[0,2],\"n\":5,\"right\":[1,3,4],\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,2,4],[0,2],[0]]}"},"fields":{"title":"A Separation Must Meet Every Condition on the Actual Space","stem":"Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 2], [0, 2, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Proposed left=[0, 2], right=[1, 3, 4]. Give (left,right,valid_separation?,classification), with classification=disconnected_separation or not_a_separation.","answer":{"kind":"expression","value":"([0, 2], [1, 3, 4], No, not_a_separation)","explanation":"Check openness,nonemptiness,disjointness and full coverage together. A valid proposal witnesses disconnection. An invalid proposal is merely not a separation; it does not establish connectedness of the whole space. The exact result, retaining the requested order, is ([0, 2], [1, 3, 4], No, not_a_separation)."},"skills":["A Separation Must Meet Every Condition on the Actual Space"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r05-connected-topological-space-0012","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"34b8679382d25ae048873e410059121b7e4e32cb309a41878216209998504fee","family_id":"qf-mod34-connected-separation-audit","seed":202611035012,"semantic_template":"qf-mod34-connected-separation-audit/{\"left\":[0,2],\"n\":5,\"right\":[1,3,4],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,4],[0,2],[0]]}"},"fields":{"title":"A Separation Must Meet Every Condition on the Actual Space","stem":"Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 2], [0, 2, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Proposed left=[0, 2], right=[1, 3, 4]. Give (left,right,valid_separation?,classification), with classification=disconnected_separation or not_a_separation.","answer":{"kind":"expression","value":"([0, 2], [1, 3, 4], No, not_a_separation)","explanation":"Check openness,nonemptiness,disjointness and full coverage together. A valid proposal witnesses disconnection. An invalid proposal is merely not a separation; it does not establish connectedness of the whole space. The exact result, retaining the requested order, is ([0, 2], [1, 3, 4], No, not_a_separation)."},"skills":["A Separation Must Meet Every Condition on the Actual Space"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r05-connected-topological-space-0013","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"db4322fea552e6e909ba46adadb0bb2447fe9dfb55d7c90c661c7f6da3e61075","family_id":"qf-mod34-connected-separation-audit","seed":202611035013,"semantic_template":"qf-mod34-connected-separation-audit/{\"left\":[0,3],\"n\":5,\"right\":[1,2,4],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,3],[0,3],[0]]}"},"fields":{"title":"A Separation Must Meet Every Condition on the Actual Space","stem":"Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 3], [0, 1, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Proposed left=[0, 3], right=[1, 2, 4]. Give (left,right,valid_separation?,classification), with classification=disconnected_separation or not_a_separation.","answer":{"kind":"expression","value":"([0, 3], [1, 2, 4], No, not_a_separation)","explanation":"Check openness,nonemptiness,disjointness and full coverage together. A valid proposal witnesses disconnection. An invalid proposal is merely not a separation; it does not establish connectedness of the whole space. The exact result, retaining the requested order, is ([0, 3], [1, 2, 4], No, not_a_separation)."},"skills":["A Separation Must Meet Every Condition on the Actual Space"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r05-connected-topological-space-0014","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1a63be0951a20bf8f459665f2ac947e77361731c232a70829d251dbf5db2944d","family_id":"qf-mod34-connected-separation-audit","seed":202611035014,"semantic_template":"qf-mod34-connected-separation-audit/{\"left\":[0,3],\"n\":5,\"right\":[1,2,4],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,3],[0,3],[0]]}"},"fields":{"title":"A Separation Must Meet Every Condition on the Actual Space","stem":"Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 3], [0, 1, 3], [0, 1, 3, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Proposed left=[0, 3], right=[1, 2, 4]. Give (left,right,valid_separation?,classification), with classification=disconnected_separation or not_a_separation.","answer":{"kind":"expression","value":"([0, 3], [1, 2, 4], No, not_a_separation)","explanation":"Check openness,nonemptiness,disjointness and full coverage together. A valid proposal witnesses disconnection. An invalid proposal is merely not a separation; it does not establish connectedness of the whole space. The exact result, retaining the requested order, is ([0, 3], [1, 2, 4], No, not_a_separation)."},"skills":["A Separation Must Meet Every Condition on the Actual Space"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r05-connected-topological-space-0015","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"61c7a78aa87722b92e55a5e11a8ea75032a9d6f2d9b3397e23d57aedc56ea5df","family_id":"qf-mod34-connected-separation-audit","seed":202611035015,"semantic_template":"qf-mod34-connected-separation-audit/{\"left\":[0,3],\"n\":5,\"right\":[1,2,4],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,2,3],[0,3],[0]]}"},"fields":{"title":"A Separation Must Meet Every Condition on the Actual Space","stem":"Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 3], [0, 2, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Proposed left=[0, 3], right=[1, 2, 4]. Give (left,right,valid_separation?,classification), with classification=disconnected_separation or not_a_separation.","answer":{"kind":"expression","value":"([0, 3], [1, 2, 4], No, not_a_separation)","explanation":"Check openness,nonemptiness,disjointness and full coverage together. A valid proposal witnesses disconnection. An invalid proposal is merely not a separation; it does not establish connectedness of the whole space. The exact result, retaining the requested order, is ([0, 3], [1, 2, 4], No, not_a_separation)."},"skills":["A Separation Must Meet Every Condition on the Actual Space"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r05-connected-topological-space-0016","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f92478273b0c785b3f2a0fe9cf6cda1b7cbc5c513853bd188fb25a9125343fed","family_id":"qf-mod34-connected-separation-audit","seed":202611035016,"semantic_template":"qf-mod34-connected-separation-audit/{\"left\":[0,3],\"n\":5,\"right\":[1,2,4],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,3],[0,3],[0]]}"},"fields":{"title":"A Separation Must Meet Every Condition on the Actual Space","stem":"Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 3], [0, 2, 3], [0, 2, 3, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Proposed left=[0, 3], right=[1, 2, 4]. Give (left,right,valid_separation?,classification), with classification=disconnected_separation or not_a_separation.","answer":{"kind":"expression","value":"([0, 3], [1, 2, 4], No, not_a_separation)","explanation":"Check openness,nonemptiness,disjointness and full coverage together. A valid proposal witnesses disconnection. An invalid proposal is merely not a separation; it does not establish connectedness of the whole space. The exact result, retaining the requested order, is ([0, 3], [1, 2, 4], No, not_a_separation)."},"skills":["A Separation Must Meet Every Condition on the Actual Space"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r05-connected-topological-space-0017","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d80562d18944b8dc0514573582c3d34b9119aebc9c8d339ebce889ffac108911","family_id":"qf-mod34-connected-separation-audit","seed":202611035017,"semantic_template":"qf-mod34-connected-separation-audit/{\"left\":[0,3],\"n\":5,\"right\":[1,2,4],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,3,4],[0,3],[0]]}"},"fields":{"title":"A Separation Must Meet Every Condition on the Actual Space","stem":"Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 3], [0, 3, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Proposed left=[0, 3], right=[1, 2, 4]. Give (left,right,valid_separation?,classification), with classification=disconnected_separation or not_a_separation.","answer":{"kind":"expression","value":"([0, 3], [1, 2, 4], No, not_a_separation)","explanation":"Check openness,nonemptiness,disjointness and full coverage together. A valid proposal witnesses disconnection. An invalid proposal is merely not a separation; it does not establish connectedness of the whole space. The exact result, retaining the requested order, is ([0, 3], [1, 2, 4], No, not_a_separation)."},"skills":["A Separation Must Meet Every Condition on the Actual Space"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r05-connected-topological-space-0018","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5170a4cc6cb8c6757a795a56f9b67565bf92fe822357c24d72db8330693bd6a2","family_id":"qf-mod34-connected-separation-audit","seed":202611035018,"semantic_template":"qf-mod34-connected-separation-audit/{\"left\":[0,3],\"n\":5,\"right\":[1,2,4],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,3,4],[0,3],[0]]}"},"fields":{"title":"A Separation Must Meet Every Condition on the Actual Space","stem":"Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 3], [0, 3, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Proposed left=[0, 3], right=[1, 2, 4]. Give (left,right,valid_separation?,classification), with classification=disconnected_separation or not_a_separation.","answer":{"kind":"expression","value":"([0, 3], [1, 2, 4], No, not_a_separation)","explanation":"Check openness,nonemptiness,disjointness and full coverage together. A valid proposal witnesses disconnection. An invalid proposal is merely not a separation; it does not establish connectedness of the whole space. The exact result, retaining the requested order, is ([0, 3], [1, 2, 4], No, not_a_separation)."},"skills":["A Separation Must Meet Every Condition on the Actual Space"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r05-connected-topological-space-0019","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"90e23f0181f66b9b0a76c431cfaf894c5893f0f784c5e79f21fea21b8a182226","family_id":"qf-mod34-connected-separation-audit","seed":202611035019,"semantic_template":"qf-mod34-connected-separation-audit/{\"left\":[0,4],\"n\":5,\"right\":[1,2,3],\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,4],[0,4],[0]]}"},"fields":{"title":"A Separation Must Meet Every Condition on the Actual Space","stem":"Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 4], [0, 1, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Proposed left=[0, 4], right=[1, 2, 3]. Give (left,right,valid_separation?,classification), with classification=disconnected_separation or not_a_separation.","answer":{"kind":"expression","value":"([0, 4], [1, 2, 3], No, not_a_separation)","explanation":"Check openness,nonemptiness,disjointness and full coverage together. A valid proposal witnesses disconnection. An invalid proposal is merely not a separation; it does not establish connectedness of the whole space. The exact result, retaining the requested order, is ([0, 4], [1, 2, 3], No, not_a_separation)."},"skills":["A Separation Must Meet Every Condition on the Actual Space"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r05-connected-topological-space-0020","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f958c28ccdc7b82d56c0684d42f3e76e3aa86c8254262d6d2ebcf7e1524c7676","family_id":"qf-mod34-connected-separation-audit","seed":202611035020,"semantic_template":"qf-mod34-connected-separation-audit/{\"left\":[0,4],\"n\":5,\"right\":[1,2,3],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,4],[0,4],[0]]}"},"fields":{"title":"A Separation Must Meet Every Condition on the Actual Space","stem":"Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 4], [0, 1, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Proposed left=[0, 4], right=[1, 2, 3]. Give (left,right,valid_separation?,classification), with classification=disconnected_separation or not_a_separation.","answer":{"kind":"expression","value":"([0, 4], [1, 2, 3], No, not_a_separation)","explanation":"Check openness,nonemptiness,disjointness and full coverage together. A valid proposal witnesses disconnection. An invalid proposal is merely not a separation; it does not establish connectedness of the whole space. The exact result, retaining the requested order, is ([0, 4], [1, 2, 3], No, not_a_separation)."},"skills":["A Separation Must Meet Every Condition on the Actual Space"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r05-connected-topological-space-0021","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fef1dd09938c6447edbd0bef0db714ee70ab346b42b1e9b9ddc5b913b2f97680","family_id":"qf-mod34-connected-separation-audit","seed":202611035021,"semantic_template":"qf-mod34-connected-separation-audit/{\"left\":[0,4],\"n\":5,\"right\":[1,2,3],\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,2,4],[0,4],[0]]}"},"fields":{"title":"A Separation Must Meet Every Condition on the Actual Space","stem":"Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 4], [0, 2, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Proposed left=[0, 4], right=[1, 2, 3]. Give (left,right,valid_separation?,classification), with classification=disconnected_separation or not_a_separation.","answer":{"kind":"expression","value":"([0, 4], [1, 2, 3], No, not_a_separation)","explanation":"Check openness,nonemptiness,disjointness and full coverage together. A valid proposal witnesses disconnection. An invalid proposal is merely not a separation; it does not establish connectedness of the whole space. The exact result, retaining the requested order, is ([0, 4], [1, 2, 3], No, not_a_separation)."},"skills":["A Separation Must Meet Every Condition on the Actual Space"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r05-connected-topological-space-0022","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"be41944041f5c77005420804344c2cde86f456cc0e467a9b65cf01b582cf28ae","family_id":"qf-mod34-connected-separation-audit","seed":202611035022,"semantic_template":"qf-mod34-connected-separation-audit/{\"left\":[0,4],\"n\":5,\"right\":[1,2,3],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,4],[0,4],[0]]}"},"fields":{"title":"A Separation Must Meet Every Condition on the Actual Space","stem":"Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 4], [0, 2, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Proposed left=[0, 4], right=[1, 2, 3]. Give (left,right,valid_separation?,classification), with classification=disconnected_separation or not_a_separation.","answer":{"kind":"expression","value":"([0, 4], [1, 2, 3], No, not_a_separation)","explanation":"Check openness,nonemptiness,disjointness and full coverage together. A valid proposal witnesses disconnection. An invalid proposal is merely not a separation; it does not establish connectedness of the whole space. The exact result, retaining the requested order, is ([0, 4], [1, 2, 3], No, not_a_separation)."},"skills":["A Separation Must Meet Every Condition on the Actual Space"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r05-connected-topological-space-0023","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7eb3055b0919170dd201da120012259527837e86250b239aaaad25e0b1b9c4ec","family_id":"qf-mod34-connected-separation-audit","seed":202611035023,"semantic_template":"qf-mod34-connected-separation-audit/{\"left\":[0,4],\"n\":5,\"right\":[1,2,3],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,3,4],[0,4],[0]]}"},"fields":{"title":"A Separation Must Meet Every Condition on the Actual Space","stem":"Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 4], [0, 3, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Proposed left=[0, 4], right=[1, 2, 3]. Give (left,right,valid_separation?,classification), with classification=disconnected_separation or not_a_separation.","answer":{"kind":"expression","value":"([0, 4], [1, 2, 3], No, not_a_separation)","explanation":"Check openness,nonemptiness,disjointness and full coverage together. A valid proposal witnesses disconnection. An invalid proposal is merely not a separation; it does not establish connectedness of the whole space. The exact result, retaining the requested order, is ([0, 4], [1, 2, 3], No, not_a_separation)."},"skills":["A Separation Must Meet Every Condition on the Actual Space"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r05-connected-topological-space-0024","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1bf6364c9cd3465786def01d1230e52e8b0050ca3ff877314d838d89234a4add","family_id":"qf-mod34-connected-separation-audit","seed":202611035024,"semantic_template":"qf-mod34-connected-separation-audit/{\"left\":[0,4],\"n\":5,\"right\":[1,2,3],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,3,4],[0,4],[0]]}"},"fields":{"title":"A Separation Must Meet Every Condition on the Actual Space","stem":"Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 4], [0, 3, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Proposed left=[0, 4], right=[1, 2, 3]. Give (left,right,valid_separation?,classification), with classification=disconnected_separation or not_a_separation.","answer":{"kind":"expression","value":"([0, 4], [1, 2, 3], No, not_a_separation)","explanation":"Check openness,nonemptiness,disjointness and full coverage together. A valid proposal witnesses disconnection. An invalid proposal is merely not a separation; it does not establish connectedness of the whole space. The exact result, retaining the requested order, is ([0, 4], [1, 2, 3], No, not_a_separation)."},"skills":["A Separation Must Meet Every Condition on the Actual Space"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r05-connected-topological-space-0025","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"006b5e5ccbf666a138882f6b718f7ac359c542378a479866f24028d50689c913","family_id":"qf-mod34-connected-separation-audit","seed":202611035025,"semantic_template":"qf-mod34-connected-separation-audit/{\"left\":[0,1],\"n\":5,\"right\":[2,3,4],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2],[0,1],[1]]}"},"fields":{"title":"A Separation Must Meet Every Condition on the Actual Space","stem":"Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [1], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Proposed left=[0, 1], right=[2, 3, 4]. Give (left,right,valid_separation?,classification), with classification=disconnected_separation or not_a_separation.","answer":{"kind":"expression","value":"([0, 1], [2, 3, 4], No, not_a_separation)","explanation":"Check openness,nonemptiness,disjointness and full coverage together. A valid proposal witnesses disconnection. An invalid proposal is merely not a separation; it does not establish connectedness of the whole space. The exact result, retaining the requested order, is ([0, 1], [2, 3, 4], No, not_a_separation)."},"skills":["A Separation Must Meet Every Condition on the Actual Space"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r05-connected-topological-space-0026","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"76552383a637f5a3860545299a8cf6826a0c5fa10adab74c43ffa0c674fadae7","family_id":"qf-mod34-connected-clopen-witness","seed":202611035026,"semantic_template":"qf-mod34-connected-clopen-witness/{\"n\":5,\"subset\":[0,1],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2],[0,1],[0]]}"},"fields":{"title":"A Clopen Witness Must Also Be Nontrivial","stem":"Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Selected subset=[0, 1]. Give (subset,complement,clopen?,nontrivial?,disconnects?,nontrivial_clopen_witness).","answer":{"kind":"expression","value":"([0, 1], [2, 3, 4], No, Yes, No, nontrivial_clopen_witness)","explanation":"Use the complement inside the actual carrier. Both sets must be open,and the selected subset must be neither empty nor whole. The disconnects flag conjoins these conditions; a failed selected witness is not an exhaustive connectedness proof. The exact result, retaining the requested order, is ([0, 1], [2, 3, 4], No, Yes, No, nontrivial_clopen_witness)."},"skills":["A Clopen Witness Must Also Be Nontrivial"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r05-connected-topological-space-0027","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bdbd48cba382e6529fb7c7120c2f43a89137ecb61c263b211d34bf24d5897553","family_id":"qf-mod34-connected-clopen-witness","seed":202611035027,"semantic_template":"qf-mod34-connected-clopen-witness/{\"n\":5,\"subset\":[0,1],\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,2],[0,1],[0]]}"},"fields":{"title":"A Clopen Witness Must Also Be Nontrivial","stem":"Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Selected subset=[0, 1]. Give (subset,complement,clopen?,nontrivial?,disconnects?,nontrivial_clopen_witness).","answer":{"kind":"expression","value":"([0, 1], [2, 3, 4], No, Yes, No, nontrivial_clopen_witness)","explanation":"Use the complement inside the actual carrier. Both sets must be open,and the selected subset must be neither empty nor whole. The disconnects flag conjoins these conditions; a failed selected witness is not an exhaustive connectedness proof. The exact result, retaining the requested order, is ([0, 1], [2, 3, 4], No, Yes, No, nontrivial_clopen_witness)."},"skills":["A Clopen Witness Must Also Be Nontrivial"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r05-connected-topological-space-0028","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6f773f7f7a6937466dbeff05810092ae39635c0fbd805241be6fc600c2516fb7","family_id":"qf-mod34-connected-clopen-witness","seed":202611035028,"semantic_template":"qf-mod34-connected-clopen-witness/{\"n\":5,\"subset\":[0,1],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,3],[0,1],[0]]}"},"fields":{"title":"A Clopen Witness Must Also Be Nontrivial","stem":"Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 1], [0, 1, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Selected subset=[0, 1]. Give (subset,complement,clopen?,nontrivial?,disconnects?,nontrivial_clopen_witness).","answer":{"kind":"expression","value":"([0, 1], [2, 3, 4], No, Yes, No, nontrivial_clopen_witness)","explanation":"Use the complement inside the actual carrier. Both sets must be open,and the selected subset must be neither empty nor whole. The disconnects flag conjoins these conditions; a failed selected witness is not an exhaustive connectedness proof. The exact result, retaining the requested order, is ([0, 1], [2, 3, 4], No, Yes, No, nontrivial_clopen_witness)."},"skills":["A Clopen Witness Must Also Be Nontrivial"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r05-connected-topological-space-0029","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4395a73ad323fb9c66fe7d03f0702a2091fafb215c7e9c50586e1e7b57bd3fe9","family_id":"qf-mod34-connected-clopen-witness","seed":202611035029,"semantic_template":"qf-mod34-connected-clopen-witness/{\"n\":5,\"subset\":[0,1],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,3],[0,1],[0]]}"},"fields":{"title":"A Clopen Witness Must Also Be Nontrivial","stem":"Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 1], [0, 1, 3], [0, 1, 3, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Selected subset=[0, 1]. Give (subset,complement,clopen?,nontrivial?,disconnects?,nontrivial_clopen_witness).","answer":{"kind":"expression","value":"([0, 1], [2, 3, 4], No, Yes, No, nontrivial_clopen_witness)","explanation":"Use the complement inside the actual carrier. Both sets must be open,and the selected subset must be neither empty nor whole. The disconnects flag conjoins these conditions; a failed selected witness is not an exhaustive connectedness proof. The exact result, retaining the requested order, is ([0, 1], [2, 3, 4], No, Yes, No, nontrivial_clopen_witness)."},"skills":["A Clopen Witness Must Also Be Nontrivial"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r05-connected-topological-space-0030","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ae604957753ad8bfc53d3da69725f1f2111b41bcf3cfcbc98ec05369cc52ed2f","family_id":"qf-mod34-connected-clopen-witness","seed":202611035030,"semantic_template":"qf-mod34-connected-clopen-witness/{\"n\":5,\"subset\":[0,1],\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,4],[0,1],[0]]}"},"fields":{"title":"A Clopen Witness Must Also Be Nontrivial","stem":"Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 1], [0, 1, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Selected subset=[0, 1]. Give (subset,complement,clopen?,nontrivial?,disconnects?,nontrivial_clopen_witness).","answer":{"kind":"expression","value":"([0, 1], [2, 3, 4], No, Yes, No, nontrivial_clopen_witness)","explanation":"Use the complement inside the actual carrier. Both sets must be open,and the selected subset must be neither empty nor whole. The disconnects flag conjoins these conditions; a failed selected witness is not an exhaustive connectedness proof. The exact result, retaining the requested order, is ([0, 1], [2, 3, 4], No, Yes, No, nontrivial_clopen_witness)."},"skills":["A Clopen Witness Must Also Be Nontrivial"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r05-connected-topological-space-0031","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3af6273cda144149949554c4ab0ed49c8e84ac866c5204eeb22166c87ca6be49","family_id":"qf-mod34-connected-clopen-witness","seed":202611035031,"semantic_template":"qf-mod34-connected-clopen-witness/{\"n\":5,\"subset\":[0,1],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,4],[0,1],[0]]}"},"fields":{"title":"A Clopen Witness Must Also Be Nontrivial","stem":"Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 1], [0, 1, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Selected subset=[0, 1]. Give (subset,complement,clopen?,nontrivial?,disconnects?,nontrivial_clopen_witness).","answer":{"kind":"expression","value":"([0, 1], [2, 3, 4], No, Yes, No, nontrivial_clopen_witness)","explanation":"Use the complement inside the actual carrier. Both sets must be open,and the selected subset must be neither empty nor whole. The disconnects flag conjoins these conditions; a failed selected witness is not an exhaustive connectedness proof. The exact result, retaining the requested order, is ([0, 1], [2, 3, 4], No, Yes, No, nontrivial_clopen_witness)."},"skills":["A Clopen Witness Must Also Be Nontrivial"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r05-connected-topological-space-0032","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"73fc9ecf4de5fe9c9f3cb535273f77b771fbf5b264b58428c9f05e2dd09fadb7","family_id":"qf-mod34-connected-clopen-witness","seed":202611035032,"semantic_template":"qf-mod34-connected-clopen-witness/{\"n\":5,\"subset\":[0,2],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2],[0,2],[0]]}"},"fields":{"title":"A Clopen Witness Must Also Be Nontrivial","stem":"Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 2], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Selected subset=[0, 2]. Give (subset,complement,clopen?,nontrivial?,disconnects?,nontrivial_clopen_witness).","answer":{"kind":"expression","value":"([0, 2], [1, 3, 4], No, Yes, No, nontrivial_clopen_witness)","explanation":"Use the complement inside the actual carrier. Both sets must be open,and the selected subset must be neither empty nor whole. The disconnects flag conjoins these conditions; a failed selected witness is not an exhaustive connectedness proof. The exact result, retaining the requested order, is ([0, 2], [1, 3, 4], No, Yes, No, nontrivial_clopen_witness)."},"skills":["A Clopen Witness Must Also Be Nontrivial"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r05-connected-topological-space-0033","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a30647b3dc7b3dd159bfde7fab617bb18511f64a54285928db5f3995d7286b5d","family_id":"qf-mod34-connected-clopen-witness","seed":202611035033,"semantic_template":"qf-mod34-connected-clopen-witness/{\"n\":5,\"subset\":[0,2],\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,2],[0,2],[0]]}"},"fields":{"title":"A Clopen Witness Must Also Be Nontrivial","stem":"Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 2], [0, 1, 2], [0, 1, 2, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Selected subset=[0, 2]. Give (subset,complement,clopen?,nontrivial?,disconnects?,nontrivial_clopen_witness).","answer":{"kind":"expression","value":"([0, 2], [1, 3, 4], No, Yes, No, nontrivial_clopen_witness)","explanation":"Use the complement inside the actual carrier. Both sets must be open,and the selected subset must be neither empty nor whole. The disconnects flag conjoins these conditions; a failed selected witness is not an exhaustive connectedness proof. The exact result, retaining the requested order, is ([0, 2], [1, 3, 4], No, Yes, No, nontrivial_clopen_witness)."},"skills":["A Clopen Witness Must Also Be Nontrivial"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r05-connected-topological-space-0034","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d329b94d145bc6f8006d289d156237f4a282d1504db16f2933a0e16ba618aa0a","family_id":"qf-mod34-connected-clopen-witness","seed":202611035034,"semantic_template":"qf-mod34-connected-clopen-witness/{\"n\":5,\"subset\":[0,2],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,2,3],[0,2],[0]]}"},"fields":{"title":"A Clopen Witness Must Also Be Nontrivial","stem":"Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 2], [0, 2, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Selected subset=[0, 2]. Give (subset,complement,clopen?,nontrivial?,disconnects?,nontrivial_clopen_witness).","answer":{"kind":"expression","value":"([0, 2], [1, 3, 4], No, Yes, No, nontrivial_clopen_witness)","explanation":"Use the complement inside the actual carrier. Both sets must be open,and the selected subset must be neither empty nor whole. The disconnects flag conjoins these conditions; a failed selected witness is not an exhaustive connectedness proof. The exact result, retaining the requested order, is ([0, 2], [1, 3, 4], No, Yes, No, nontrivial_clopen_witness)."},"skills":["A Clopen Witness Must Also Be Nontrivial"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r05-connected-topological-space-0035","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"83af65c6608a0bcef0271ff8b970e3f7d4311876168ae700fbc2832f94f24eaa","family_id":"qf-mod34-connected-clopen-witness","seed":202611035035,"semantic_template":"qf-mod34-connected-clopen-witness/{\"n\":5,\"subset\":[0,2],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,3],[0,2],[0]]}"},"fields":{"title":"A Clopen Witness Must Also Be Nontrivial","stem":"Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Selected subset=[0, 2]. Give (subset,complement,clopen?,nontrivial?,disconnects?,nontrivial_clopen_witness).","answer":{"kind":"expression","value":"([0, 2], [1, 3, 4], No, Yes, No, nontrivial_clopen_witness)","explanation":"Use the complement inside the actual carrier. Both sets must be open,and the selected subset must be neither empty nor whole. The disconnects flag conjoins these conditions; a failed selected witness is not an exhaustive connectedness proof. The exact result, retaining the requested order, is ([0, 2], [1, 3, 4], No, Yes, No, nontrivial_clopen_witness)."},"skills":["A Clopen Witness Must Also Be Nontrivial"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r05-connected-topological-space-0036","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"91ab67c78fc349a1d4b9278de18666c52d2e0f353c19205c123e1221db6d2189","family_id":"qf-mod34-connected-clopen-witness","seed":202611035036,"semantic_template":"qf-mod34-connected-clopen-witness/{\"n\":5,\"subset\":[0,2],\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,2,4],[0,2],[0]]}"},"fields":{"title":"A Clopen Witness Must Also Be Nontrivial","stem":"Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 2], [0, 2, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Selected subset=[0, 2]. Give (subset,complement,clopen?,nontrivial?,disconnects?,nontrivial_clopen_witness).","answer":{"kind":"expression","value":"([0, 2], [1, 3, 4], No, Yes, No, nontrivial_clopen_witness)","explanation":"Use the complement inside the actual carrier. Both sets must be open,and the selected subset must be neither empty nor whole. The disconnects flag conjoins these conditions; a failed selected witness is not an exhaustive connectedness proof. The exact result, retaining the requested order, is ([0, 2], [1, 3, 4], No, Yes, No, nontrivial_clopen_witness)."},"skills":["A Clopen Witness Must Also Be Nontrivial"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r05-connected-topological-space-0037","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"00e0bf3dd3bfa6947f2b140fee04de39616e3af13cb4fbf12a3edce474aa1e5c","family_id":"qf-mod34-connected-clopen-witness","seed":202611035037,"semantic_template":"qf-mod34-connected-clopen-witness/{\"n\":5,\"subset\":[0,2],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,4],[0,2],[0]]}"},"fields":{"title":"A Clopen Witness Must Also Be Nontrivial","stem":"Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 2], [0, 2, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Selected subset=[0, 2]. Give (subset,complement,clopen?,nontrivial?,disconnects?,nontrivial_clopen_witness).","answer":{"kind":"expression","value":"([0, 2], [1, 3, 4], No, Yes, No, nontrivial_clopen_witness)","explanation":"Use the complement inside the actual carrier. Both sets must be open,and the selected subset must be neither empty nor whole. The disconnects flag conjoins these conditions; a failed selected witness is not an exhaustive connectedness proof. The exact result, retaining the requested order, is ([0, 2], [1, 3, 4], No, Yes, No, nontrivial_clopen_witness)."},"skills":["A Clopen Witness Must Also Be Nontrivial"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r05-connected-topological-space-0038","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"653e76a644a908072d30a659ed2239821c4cabc8fe32bd1c4397e019bbb1cdd0","family_id":"qf-mod34-connected-clopen-witness","seed":202611035038,"semantic_template":"qf-mod34-connected-clopen-witness/{\"n\":5,\"subset\":[0,3],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,3],[0,3],[0]]}"},"fields":{"title":"A Clopen Witness Must Also Be Nontrivial","stem":"Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 3], [0, 1, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Selected subset=[0, 3]. Give (subset,complement,clopen?,nontrivial?,disconnects?,nontrivial_clopen_witness).","answer":{"kind":"expression","value":"([0, 3], [1, 2, 4], No, Yes, No, nontrivial_clopen_witness)","explanation":"Use the complement inside the actual carrier. Both sets must be open,and the selected subset must be neither empty nor whole. The disconnects flag conjoins these conditions; a failed selected witness is not an exhaustive connectedness proof. The exact result, retaining the requested order, is ([0, 3], [1, 2, 4], No, Yes, No, nontrivial_clopen_witness)."},"skills":["A Clopen Witness Must Also Be Nontrivial"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r05-connected-topological-space-0039","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f165c4f1ff0c4dcd1736a0cc74b6e1925558f44c0b81c1f0ea990c3c8389a524","family_id":"qf-mod34-connected-clopen-witness","seed":202611035039,"semantic_template":"qf-mod34-connected-clopen-witness/{\"n\":5,\"subset\":[0,3],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,3],[0,3],[0]]}"},"fields":{"title":"A Clopen Witness Must Also Be Nontrivial","stem":"Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 3], [0, 1, 3], [0, 1, 3, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Selected subset=[0, 3]. Give (subset,complement,clopen?,nontrivial?,disconnects?,nontrivial_clopen_witness).","answer":{"kind":"expression","value":"([0, 3], [1, 2, 4], No, Yes, No, nontrivial_clopen_witness)","explanation":"Use the complement inside the actual carrier. Both sets must be open,and the selected subset must be neither empty nor whole. The disconnects flag conjoins these conditions; a failed selected witness is not an exhaustive connectedness proof. The exact result, retaining the requested order, is ([0, 3], [1, 2, 4], No, Yes, No, nontrivial_clopen_witness)."},"skills":["A Clopen Witness Must Also Be Nontrivial"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r05-connected-topological-space-0040","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6f03a53eca81f388dea62dcfd915f12a2d2fc2a6e5b634fac395a2e6d886a71a","family_id":"qf-mod34-connected-clopen-witness","seed":202611035040,"semantic_template":"qf-mod34-connected-clopen-witness/{\"n\":5,\"subset\":[0,3],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,2,3],[0,3],[0]]}"},"fields":{"title":"A Clopen Witness Must Also Be Nontrivial","stem":"Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 3], [0, 2, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Selected subset=[0, 3]. Give (subset,complement,clopen?,nontrivial?,disconnects?,nontrivial_clopen_witness).","answer":{"kind":"expression","value":"([0, 3], [1, 2, 4], No, Yes, No, nontrivial_clopen_witness)","explanation":"Use the complement inside the actual carrier. Both sets must be open,and the selected subset must be neither empty nor whole. The disconnects flag conjoins these conditions; a failed selected witness is not an exhaustive connectedness proof. The exact result, retaining the requested order, is ([0, 3], [1, 2, 4], No, Yes, No, nontrivial_clopen_witness)."},"skills":["A Clopen Witness Must Also Be Nontrivial"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r05-connected-topological-space-0041","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7e2052a2dacffd30238b64ac4af7f122f8f9d16a0b21d1f510dabee4489867eb","family_id":"qf-mod34-connected-clopen-witness","seed":202611035041,"semantic_template":"qf-mod34-connected-clopen-witness/{\"n\":5,\"subset\":[0,3],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,3],[0,3],[0]]}"},"fields":{"title":"A Clopen Witness Must Also Be Nontrivial","stem":"Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 3], [0, 2, 3], [0, 2, 3, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Selected subset=[0, 3]. Give (subset,complement,clopen?,nontrivial?,disconnects?,nontrivial_clopen_witness).","answer":{"kind":"expression","value":"([0, 3], [1, 2, 4], No, Yes, No, nontrivial_clopen_witness)","explanation":"Use the complement inside the actual carrier. Both sets must be open,and the selected subset must be neither empty nor whole. The disconnects flag conjoins these conditions; a failed selected witness is not an exhaustive connectedness proof. The exact result, retaining the requested order, is ([0, 3], [1, 2, 4], No, Yes, No, nontrivial_clopen_witness)."},"skills":["A Clopen Witness Must Also Be Nontrivial"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r05-connected-topological-space-0042","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dd67620dc0047714aa6147f779845994147df19660fd7aa0d31b21ade6debd4c","family_id":"qf-mod34-connected-clopen-witness","seed":202611035042,"semantic_template":"qf-mod34-connected-clopen-witness/{\"n\":5,\"subset\":[0,3],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,3,4],[0,3],[0]]}"},"fields":{"title":"A Clopen Witness Must Also Be Nontrivial","stem":"Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 3], [0, 3, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Selected subset=[0, 3]. Give (subset,complement,clopen?,nontrivial?,disconnects?,nontrivial_clopen_witness).","answer":{"kind":"expression","value":"([0, 3], [1, 2, 4], No, Yes, No, nontrivial_clopen_witness)","explanation":"Use the complement inside the actual carrier. Both sets must be open,and the selected subset must be neither empty nor whole. The disconnects flag conjoins these conditions; a failed selected witness is not an exhaustive connectedness proof. The exact result, retaining the requested order, is ([0, 3], [1, 2, 4], No, Yes, No, nontrivial_clopen_witness)."},"skills":["A Clopen Witness Must Also Be Nontrivial"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r05-connected-topological-space-0043","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fe9273d4b7ba1f7081a0aeabf383c4cac96f7da77055fff0432e35b70ab91556","family_id":"qf-mod34-connected-clopen-witness","seed":202611035043,"semantic_template":"qf-mod34-connected-clopen-witness/{\"n\":5,\"subset\":[0,3],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,3,4],[0,3],[0]]}"},"fields":{"title":"A Clopen Witness Must Also Be Nontrivial","stem":"Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 3], [0, 3, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Selected subset=[0, 3]. Give (subset,complement,clopen?,nontrivial?,disconnects?,nontrivial_clopen_witness).","answer":{"kind":"expression","value":"([0, 3], [1, 2, 4], No, Yes, No, nontrivial_clopen_witness)","explanation":"Use the complement inside the actual carrier. Both sets must be open,and the selected subset must be neither empty nor whole. The disconnects flag conjoins these conditions; a failed selected witness is not an exhaustive connectedness proof. The exact result, retaining the requested order, is ([0, 3], [1, 2, 4], No, Yes, No, nontrivial_clopen_witness)."},"skills":["A Clopen Witness Must Also Be Nontrivial"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r05-connected-topological-space-0044","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1e464f136b91a1e9cc5f03454d3747178a15ab6c19ee930600f06aff9d8816f8","family_id":"qf-mod34-connected-clopen-witness","seed":202611035044,"semantic_template":"qf-mod34-connected-clopen-witness/{\"n\":5,\"subset\":[0,4],\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,4],[0,4],[0]]}"},"fields":{"title":"A Clopen Witness Must Also Be Nontrivial","stem":"Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 4], [0, 1, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Selected subset=[0, 4]. Give (subset,complement,clopen?,nontrivial?,disconnects?,nontrivial_clopen_witness).","answer":{"kind":"expression","value":"([0, 4], [1, 2, 3], No, Yes, No, nontrivial_clopen_witness)","explanation":"Use the complement inside the actual carrier. Both sets must be open,and the selected subset must be neither empty nor whole. The disconnects flag conjoins these conditions; a failed selected witness is not an exhaustive connectedness proof. The exact result, retaining the requested order, is ([0, 4], [1, 2, 3], No, Yes, No, nontrivial_clopen_witness)."},"skills":["A Clopen Witness Must Also Be Nontrivial"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r05-connected-topological-space-0045","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0a81692d15b3bdcfb45898deff42cb79f8a54a9aa35263cf348633ca0d0d9e70","family_id":"qf-mod34-connected-clopen-witness","seed":202611035045,"semantic_template":"qf-mod34-connected-clopen-witness/{\"n\":5,\"subset\":[0,4],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,4],[0,4],[0]]}"},"fields":{"title":"A Clopen Witness Must Also Be Nontrivial","stem":"Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 4], [0, 1, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Selected subset=[0, 4]. Give (subset,complement,clopen?,nontrivial?,disconnects?,nontrivial_clopen_witness).","answer":{"kind":"expression","value":"([0, 4], [1, 2, 3], No, Yes, No, nontrivial_clopen_witness)","explanation":"Use the complement inside the actual carrier. Both sets must be open,and the selected subset must be neither empty nor whole. The disconnects flag conjoins these conditions; a failed selected witness is not an exhaustive connectedness proof. The exact result, retaining the requested order, is ([0, 4], [1, 2, 3], No, Yes, No, nontrivial_clopen_witness)."},"skills":["A Clopen Witness Must Also Be Nontrivial"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r05-connected-topological-space-0046","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fe271da0acea7c8f99c23b774fbfc0264328421bf7646b669f113029014bd771","family_id":"qf-mod34-connected-clopen-witness","seed":202611035046,"semantic_template":"qf-mod34-connected-clopen-witness/{\"n\":5,\"subset\":[0,4],\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,2,4],[0,4],[0]]}"},"fields":{"title":"A Clopen Witness Must Also Be Nontrivial","stem":"Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 4], [0, 2, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Selected subset=[0, 4]. Give (subset,complement,clopen?,nontrivial?,disconnects?,nontrivial_clopen_witness).","answer":{"kind":"expression","value":"([0, 4], [1, 2, 3], No, Yes, No, nontrivial_clopen_witness)","explanation":"Use the complement inside the actual carrier. Both sets must be open,and the selected subset must be neither empty nor whole. The disconnects flag conjoins these conditions; a failed selected witness is not an exhaustive connectedness proof. The exact result, retaining the requested order, is ([0, 4], [1, 2, 3], No, Yes, No, nontrivial_clopen_witness)."},"skills":["A Clopen Witness Must Also Be Nontrivial"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r05-connected-topological-space-0047","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b7d2ce4cb7b23cb9db8bead1b2d418b2c57bdb5043ddd79ff87707351c0e2552","family_id":"qf-mod34-connected-clopen-witness","seed":202611035047,"semantic_template":"qf-mod34-connected-clopen-witness/{\"n\":5,\"subset\":[0,4],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,4],[0,4],[0]]}"},"fields":{"title":"A Clopen Witness Must Also Be Nontrivial","stem":"Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 4], [0, 2, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Selected subset=[0, 4]. Give (subset,complement,clopen?,nontrivial?,disconnects?,nontrivial_clopen_witness).","answer":{"kind":"expression","value":"([0, 4], [1, 2, 3], No, Yes, No, nontrivial_clopen_witness)","explanation":"Use the complement inside the actual carrier. Both sets must be open,and the selected subset must be neither empty nor whole. The disconnects flag conjoins these conditions; a failed selected witness is not an exhaustive connectedness proof. The exact result, retaining the requested order, is ([0, 4], [1, 2, 3], No, Yes, No, nontrivial_clopen_witness)."},"skills":["A Clopen Witness Must Also Be Nontrivial"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r05-connected-topological-space-0048","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"af662df386bd45d24cafb2496064b89ea7c7541e6d7035cb26c619bcc7b42c48","family_id":"qf-mod34-connected-clopen-witness","seed":202611035048,"semantic_template":"qf-mod34-connected-clopen-witness/{\"n\":5,\"subset\":[0,4],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,3,4],[0,4],[0]]}"},"fields":{"title":"A Clopen Witness Must Also Be Nontrivial","stem":"Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 4], [0, 3, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Selected subset=[0, 4]. Give (subset,complement,clopen?,nontrivial?,disconnects?,nontrivial_clopen_witness).","answer":{"kind":"expression","value":"([0, 4], [1, 2, 3], No, Yes, No, nontrivial_clopen_witness)","explanation":"Use the complement inside the actual carrier. Both sets must be open,and the selected subset must be neither empty nor whole. The disconnects flag conjoins these conditions; a failed selected witness is not an exhaustive connectedness proof. The exact result, retaining the requested order, is ([0, 4], [1, 2, 3], No, Yes, No, nontrivial_clopen_witness)."},"skills":["A Clopen Witness Must Also Be Nontrivial"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r05-connected-topological-space-0049","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e96c112c62135a6b3d19082c822706729f50bd75a78d31ee4ff4fcace9740a12","family_id":"qf-mod34-connected-clopen-witness","seed":202611035049,"semantic_template":"qf-mod34-connected-clopen-witness/{\"n\":5,\"subset\":[0,4],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,3,4],[0,4],[0]]}"},"fields":{"title":"A Clopen Witness Must Also Be Nontrivial","stem":"Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 4], [0, 3, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Selected subset=[0, 4]. Give (subset,complement,clopen?,nontrivial?,disconnects?,nontrivial_clopen_witness).","answer":{"kind":"expression","value":"([0, 4], [1, 2, 3], No, Yes, No, nontrivial_clopen_witness)","explanation":"Use the complement inside the actual carrier. Both sets must be open,and the selected subset must be neither empty nor whole. The disconnects flag conjoins these conditions; a failed selected witness is not an exhaustive connectedness proof. The exact result, retaining the requested order, is ([0, 4], [1, 2, 3], No, Yes, No, nontrivial_clopen_witness)."},"skills":["A Clopen Witness Must Also Be Nontrivial"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r05-connected-topological-space-0050","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a5c34e4237cd749fac4131a63b40b52c00e4376469cb804f5b467c905a7f02d5","family_id":"qf-mod34-connected-clopen-witness","seed":202611035050,"semantic_template":"qf-mod34-connected-clopen-witness/{\"n\":5,\"subset\":[0,1],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2],[0,1],[1]]}"},"fields":{"title":"A Clopen Witness Must Also Be Nontrivial","stem":"Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [1], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Selected subset=[0, 1]. Give (subset,complement,clopen?,nontrivial?,disconnects?,nontrivial_clopen_witness).","answer":{"kind":"expression","value":"([0, 1], [2, 3, 4], No, Yes, No, nontrivial_clopen_witness)","explanation":"Use the complement inside the actual carrier. Both sets must be open,and the selected subset must be neither empty nor whole. The disconnects flag conjoins these conditions; a failed selected witness is not an exhaustive connectedness proof. The exact result, retaining the requested order, is ([0, 1], [2, 3, 4], No, Yes, No, nontrivial_clopen_witness)."},"skills":["A Clopen Witness Must Also Be Nontrivial"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r05-connected-topological-space-0051","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1163dd346be215ad97aad1417aafe15253ad73a0f4f2d1a9650b90b1d6655b91","family_id":"qf-mod34-connected-exhaustive","seed":202611035051,"semantic_template":"qf-mod34-connected-exhaustive/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2],[0,1],[0]]}"},"fields":{"title":"Exhaust the Complete Finite Topology before Certifying Connectedness","stem":"Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Give (clopen_sets,nontrivial_clopen,connected?,exhaustive_complete_finite_topology), retaining the supplied topology order within both lists.","answer":{"kind":"expression","value":"([[], [0, 1, 2, 3, 4]], [], Yes, exhaustive_complete_finite_topology)","explanation":"Check the complement of every declared open subset and list every clopen set. Remove empty and whole sets to form the nontrivial list. Exactly an empty nontrivial list certifies connectedness under this explicit convention. The exact result, retaining the requested order, is ([[], [0, 1, 2, 3, 4]], [], Yes, exhaustive_complete_finite_topology)."},"skills":["Exhaust the Complete Finite Topology before Certifying Connectedness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r05-connected-topological-space-0052","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dd991fde36889f089fa20f7639dd8df7a16f627c42d6f1968893d8b32e7cbf64","family_id":"qf-mod34-connected-exhaustive","seed":202611035052,"semantic_template":"qf-mod34-connected-exhaustive/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,2],[0,1],[0]]}"},"fields":{"title":"Exhaust the Complete Finite Topology before Certifying Connectedness","stem":"Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Give (clopen_sets,nontrivial_clopen,connected?,exhaustive_complete_finite_topology), retaining the supplied topology order within both lists.","answer":{"kind":"expression","value":"([[], [0, 1, 2, 3, 4]], [], Yes, exhaustive_complete_finite_topology)","explanation":"Check the complement of every declared open subset and list every clopen set. Remove empty and whole sets to form the nontrivial list. Exactly an empty nontrivial list certifies connectedness under this explicit convention. The exact result, retaining the requested order, is ([[], [0, 1, 2, 3, 4]], [], Yes, exhaustive_complete_finite_topology)."},"skills":["Exhaust the Complete Finite Topology before Certifying Connectedness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r05-connected-topological-space-0053","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e77d411fbf44303f8f7aadb032383e674e073e6f73e58c2f356cf871e166e483","family_id":"qf-mod34-connected-exhaustive","seed":202611035053,"semantic_template":"qf-mod34-connected-exhaustive/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,3],[0,1],[0]]}"},"fields":{"title":"Exhaust the Complete Finite Topology before Certifying Connectedness","stem":"Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 1], [0, 1, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Give (clopen_sets,nontrivial_clopen,connected?,exhaustive_complete_finite_topology), retaining the supplied topology order within both lists.","answer":{"kind":"expression","value":"([[], [0, 1, 2, 3, 4]], [], Yes, exhaustive_complete_finite_topology)","explanation":"Check the complement of every declared open subset and list every clopen set. Remove empty and whole sets to form the nontrivial list. Exactly an empty nontrivial list certifies connectedness under this explicit convention. The exact result, retaining the requested order, is ([[], [0, 1, 2, 3, 4]], [], Yes, exhaustive_complete_finite_topology)."},"skills":["Exhaust the Complete Finite Topology before Certifying Connectedness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r05-connected-topological-space-0054","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"08a5036657b44b8c5aac152131dba317cf8bb972702b27461f6068f271ec6812","family_id":"qf-mod34-connected-exhaustive","seed":202611035054,"semantic_template":"qf-mod34-connected-exhaustive/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,3],[0,1],[0]]}"},"fields":{"title":"Exhaust the Complete Finite Topology before Certifying Connectedness","stem":"Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 1], [0, 1, 3], [0, 1, 3, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Give (clopen_sets,nontrivial_clopen,connected?,exhaustive_complete_finite_topology), retaining the supplied topology order within both lists.","answer":{"kind":"expression","value":"([[], [0, 1, 2, 3, 4]], [], Yes, exhaustive_complete_finite_topology)","explanation":"Check the complement of every declared open subset and list every clopen set. Remove empty and whole sets to form the nontrivial list. Exactly an empty nontrivial list certifies connectedness under this explicit convention. The exact result, retaining the requested order, is ([[], [0, 1, 2, 3, 4]], [], Yes, exhaustive_complete_finite_topology)."},"skills":["Exhaust the Complete Finite Topology before Certifying Connectedness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r05-connected-topological-space-0055","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1fddb1136c66d63ec774c3ade12638db26a79213f7e2dd557a4724d1aa09593e","family_id":"qf-mod34-connected-exhaustive","seed":202611035055,"semantic_template":"qf-mod34-connected-exhaustive/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,4],[0,1],[0]]}"},"fields":{"title":"Exhaust the Complete Finite Topology before Certifying Connectedness","stem":"Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 1], [0, 1, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Give (clopen_sets,nontrivial_clopen,connected?,exhaustive_complete_finite_topology), retaining the supplied topology order within both lists.","answer":{"kind":"expression","value":"([[], [0, 1, 2, 3, 4]], [], Yes, exhaustive_complete_finite_topology)","explanation":"Check the complement of every declared open subset and list every clopen set. Remove empty and whole sets to form the nontrivial list. Exactly an empty nontrivial list certifies connectedness under this explicit convention. The exact result, retaining the requested order, is ([[], [0, 1, 2, 3, 4]], [], Yes, exhaustive_complete_finite_topology)."},"skills":["Exhaust the Complete Finite Topology before Certifying Connectedness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r05-connected-topological-space-0056","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6ac0985e897c99ce5694b0b162e5d7b473c7f08d931ae2c873319f439ec12f8f","family_id":"qf-mod34-connected-exhaustive","seed":202611035056,"semantic_template":"qf-mod34-connected-exhaustive/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,4],[0,1],[0]]}"},"fields":{"title":"Exhaust the Complete Finite Topology before Certifying Connectedness","stem":"Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 1], [0, 1, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Give (clopen_sets,nontrivial_clopen,connected?,exhaustive_complete_finite_topology), retaining the supplied topology order within both lists.","answer":{"kind":"expression","value":"([[], [0, 1, 2, 3, 4]], [], Yes, exhaustive_complete_finite_topology)","explanation":"Check the complement of every declared open subset and list every clopen set. Remove empty and whole sets to form the nontrivial list. Exactly an empty nontrivial list certifies connectedness under this explicit convention. The exact result, retaining the requested order, is ([[], [0, 1, 2, 3, 4]], [], Yes, exhaustive_complete_finite_topology)."},"skills":["Exhaust the Complete Finite Topology before Certifying Connectedness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r05-connected-topological-space-0057","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5a91fa2b2b4ffb53edbfa638a4775c6a1bc98aeb867b17a78a9e902d4efd17c7","family_id":"qf-mod34-connected-exhaustive","seed":202611035057,"semantic_template":"qf-mod34-connected-exhaustive/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2],[0,2],[0]]}"},"fields":{"title":"Exhaust the Complete Finite Topology before Certifying Connectedness","stem":"Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 2], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Give (clopen_sets,nontrivial_clopen,connected?,exhaustive_complete_finite_topology), retaining the supplied topology order within both lists.","answer":{"kind":"expression","value":"([[], [0, 1, 2, 3, 4]], [], Yes, exhaustive_complete_finite_topology)","explanation":"Check the complement of every declared open subset and list every clopen set. Remove empty and whole sets to form the nontrivial list. Exactly an empty nontrivial list certifies connectedness under this explicit convention. The exact result, retaining the requested order, is ([[], [0, 1, 2, 3, 4]], [], Yes, exhaustive_complete_finite_topology)."},"skills":["Exhaust the Complete Finite Topology before Certifying Connectedness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r05-connected-topological-space-0058","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2e19a7718c6b12af83c80421f04735b23a6a2277a08b726029b2d5d1b5dfc3ea","family_id":"qf-mod34-connected-exhaustive","seed":202611035058,"semantic_template":"qf-mod34-connected-exhaustive/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,2],[0,2],[0]]}"},"fields":{"title":"Exhaust the Complete Finite Topology before Certifying Connectedness","stem":"Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 2], [0, 1, 2], [0, 1, 2, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Give (clopen_sets,nontrivial_clopen,connected?,exhaustive_complete_finite_topology), retaining the supplied topology order within both lists.","answer":{"kind":"expression","value":"([[], [0, 1, 2, 3, 4]], [], Yes, exhaustive_complete_finite_topology)","explanation":"Check the complement of every declared open subset and list every clopen set. Remove empty and whole sets to form the nontrivial list. Exactly an empty nontrivial list certifies connectedness under this explicit convention. The exact result, retaining the requested order, is ([[], [0, 1, 2, 3, 4]], [], Yes, exhaustive_complete_finite_topology)."},"skills":["Exhaust the Complete Finite Topology before Certifying Connectedness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r05-connected-topological-space-0059","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4c4faad5feb6c88fc16894fa0832a22e88b58b63556d556c96de759d63315a42","family_id":"qf-mod34-connected-exhaustive","seed":202611035059,"semantic_template":"qf-mod34-connected-exhaustive/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,2,3],[0,2],[0]]}"},"fields":{"title":"Exhaust the Complete Finite Topology before Certifying Connectedness","stem":"Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 2], [0, 2, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Give (clopen_sets,nontrivial_clopen,connected?,exhaustive_complete_finite_topology), retaining the supplied topology order within both lists.","answer":{"kind":"expression","value":"([[], [0, 1, 2, 3, 4]], [], Yes, exhaustive_complete_finite_topology)","explanation":"Check the complement of every declared open subset and list every clopen set. Remove empty and whole sets to form the nontrivial list. Exactly an empty nontrivial list certifies connectedness under this explicit convention. The exact result, retaining the requested order, is ([[], [0, 1, 2, 3, 4]], [], Yes, exhaustive_complete_finite_topology)."},"skills":["Exhaust the Complete Finite Topology before Certifying Connectedness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r05-connected-topological-space-0060","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"00689c639773bfc9ae73d48f2bec2f232277d451323621118f0dab68e040d065","family_id":"qf-mod34-connected-exhaustive","seed":202611035060,"semantic_template":"qf-mod34-connected-exhaustive/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,3],[0,2],[0]]}"},"fields":{"title":"Exhaust the Complete Finite Topology before Certifying Connectedness","stem":"Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Give (clopen_sets,nontrivial_clopen,connected?,exhaustive_complete_finite_topology), retaining the supplied topology order within both lists.","answer":{"kind":"expression","value":"([[], [0, 1, 2, 3, 4]], [], Yes, exhaustive_complete_finite_topology)","explanation":"Check the complement of every declared open subset and list every clopen set. Remove empty and whole sets to form the nontrivial list. Exactly an empty nontrivial list certifies connectedness under this explicit convention. The exact result, retaining the requested order, is ([[], [0, 1, 2, 3, 4]], [], Yes, exhaustive_complete_finite_topology)."},"skills":["Exhaust the Complete Finite Topology before Certifying Connectedness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r05-connected-topological-space-0061","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"40ca5eced7fb3d136e87ff23ab92602aea94ba9b6b166580ae6ad990001f1cd6","family_id":"qf-mod34-connected-exhaustive","seed":202611035061,"semantic_template":"qf-mod34-connected-exhaustive/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,2,4],[0,2],[0]]}"},"fields":{"title":"Exhaust the Complete Finite Topology before Certifying Connectedness","stem":"Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 2], [0, 2, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Give (clopen_sets,nontrivial_clopen,connected?,exhaustive_complete_finite_topology), retaining the supplied topology order within both lists.","answer":{"kind":"expression","value":"([[], [0, 1, 2, 3, 4]], [], Yes, exhaustive_complete_finite_topology)","explanation":"Check the complement of every declared open subset and list every clopen set. Remove empty and whole sets to form the nontrivial list. Exactly an empty nontrivial list certifies connectedness under this explicit convention. The exact result, retaining the requested order, is ([[], [0, 1, 2, 3, 4]], [], Yes, exhaustive_complete_finite_topology)."},"skills":["Exhaust the Complete Finite Topology before Certifying Connectedness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r05-connected-topological-space-0062","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"616faebbfb40e418a3c7c0f9016fe1874dee39964e6f9474bc8c10126e7d08aa","family_id":"qf-mod34-connected-exhaustive","seed":202611035062,"semantic_template":"qf-mod34-connected-exhaustive/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,4],[0,2],[0]]}"},"fields":{"title":"Exhaust the Complete Finite Topology before Certifying Connectedness","stem":"Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 2], [0, 2, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Give (clopen_sets,nontrivial_clopen,connected?,exhaustive_complete_finite_topology), retaining the supplied topology order within both lists.","answer":{"kind":"expression","value":"([[], [0, 1, 2, 3, 4]], [], Yes, exhaustive_complete_finite_topology)","explanation":"Check the complement of every declared open subset and list every clopen set. Remove empty and whole sets to form the nontrivial list. Exactly an empty nontrivial list certifies connectedness under this explicit convention. The exact result, retaining the requested order, is ([[], [0, 1, 2, 3, 4]], [], Yes, exhaustive_complete_finite_topology)."},"skills":["Exhaust the Complete Finite Topology before Certifying Connectedness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r05-connected-topological-space-0063","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8cb59bda361b14c6ac82676938137a98c5720f3de5126f422bae526ee41080ab","family_id":"qf-mod34-connected-exhaustive","seed":202611035063,"semantic_template":"qf-mod34-connected-exhaustive/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,3],[0,3],[0]]}"},"fields":{"title":"Exhaust the Complete Finite Topology before Certifying Connectedness","stem":"Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 3], [0, 1, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Give (clopen_sets,nontrivial_clopen,connected?,exhaustive_complete_finite_topology), retaining the supplied topology order within both lists.","answer":{"kind":"expression","value":"([[], [0, 1, 2, 3, 4]], [], Yes, exhaustive_complete_finite_topology)","explanation":"Check the complement of every declared open subset and list every clopen set. Remove empty and whole sets to form the nontrivial list. Exactly an empty nontrivial list certifies connectedness under this explicit convention. The exact result, retaining the requested order, is ([[], [0, 1, 2, 3, 4]], [], Yes, exhaustive_complete_finite_topology)."},"skills":["Exhaust the Complete Finite Topology before Certifying Connectedness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r05-connected-topological-space-0064","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"66e4f64d8105140f00605db3905d09982a612c1b8ad109af2a5e5b756eaa6053","family_id":"qf-mod34-connected-exhaustive","seed":202611035064,"semantic_template":"qf-mod34-connected-exhaustive/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,3],[0,3],[0]]}"},"fields":{"title":"Exhaust the Complete Finite Topology before Certifying Connectedness","stem":"Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 3], [0, 1, 3], [0, 1, 3, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Give (clopen_sets,nontrivial_clopen,connected?,exhaustive_complete_finite_topology), retaining the supplied topology order within both lists.","answer":{"kind":"expression","value":"([[], [0, 1, 2, 3, 4]], [], Yes, exhaustive_complete_finite_topology)","explanation":"Check the complement of every declared open subset and list every clopen set. Remove empty and whole sets to form the nontrivial list. Exactly an empty nontrivial list certifies connectedness under this explicit convention. The exact result, retaining the requested order, is ([[], [0, 1, 2, 3, 4]], [], Yes, exhaustive_complete_finite_topology)."},"skills":["Exhaust the Complete Finite Topology before Certifying Connectedness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r05-connected-topological-space-0065","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"40959c9654ac1f59d45debdbf45b2e9e3e0b44dad1eaf6f071b756c7b3677b01","family_id":"qf-mod34-connected-exhaustive","seed":202611035065,"semantic_template":"qf-mod34-connected-exhaustive/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,2,3],[0,3],[0]]}"},"fields":{"title":"Exhaust the Complete Finite Topology before Certifying Connectedness","stem":"Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 3], [0, 2, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Give (clopen_sets,nontrivial_clopen,connected?,exhaustive_complete_finite_topology), retaining the supplied topology order within both lists.","answer":{"kind":"expression","value":"([[], [0, 1, 2, 3, 4]], [], Yes, exhaustive_complete_finite_topology)","explanation":"Check the complement of every declared open subset and list every clopen set. Remove empty and whole sets to form the nontrivial list. Exactly an empty nontrivial list certifies connectedness under this explicit convention. The exact result, retaining the requested order, is ([[], [0, 1, 2, 3, 4]], [], Yes, exhaustive_complete_finite_topology)."},"skills":["Exhaust the Complete Finite Topology before Certifying Connectedness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r05-connected-topological-space-0066","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"52713bbda1db2e834be8749bff6ac8b0c7c5c3c68b3842e1748f4b459db5fffa","family_id":"qf-mod34-connected-exhaustive","seed":202611035066,"semantic_template":"qf-mod34-connected-exhaustive/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,3],[0,3],[0]]}"},"fields":{"title":"Exhaust the Complete Finite Topology before Certifying Connectedness","stem":"Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 3], [0, 2, 3], [0, 2, 3, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Give (clopen_sets,nontrivial_clopen,connected?,exhaustive_complete_finite_topology), retaining the supplied topology order within both lists.","answer":{"kind":"expression","value":"([[], [0, 1, 2, 3, 4]], [], Yes, exhaustive_complete_finite_topology)","explanation":"Check the complement of every declared open subset and list every clopen set. Remove empty and whole sets to form the nontrivial list. Exactly an empty nontrivial list certifies connectedness under this explicit convention. The exact result, retaining the requested order, is ([[], [0, 1, 2, 3, 4]], [], Yes, exhaustive_complete_finite_topology)."},"skills":["Exhaust the Complete Finite Topology before Certifying Connectedness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r05-connected-topological-space-0067","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5d419975a3d6a1344ab5950d2bb407425a6233c591ac41c6cbda734f6f96c3b5","family_id":"qf-mod34-connected-exhaustive","seed":202611035067,"semantic_template":"qf-mod34-connected-exhaustive/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,3,4],[0,3],[0]]}"},"fields":{"title":"Exhaust the Complete Finite Topology before Certifying Connectedness","stem":"Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 3], [0, 3, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Give (clopen_sets,nontrivial_clopen,connected?,exhaustive_complete_finite_topology), retaining the supplied topology order within both lists.","answer":{"kind":"expression","value":"([[], [0, 1, 2, 3, 4]], [], Yes, exhaustive_complete_finite_topology)","explanation":"Check the complement of every declared open subset and list every clopen set. Remove empty and whole sets to form the nontrivial list. Exactly an empty nontrivial list certifies connectedness under this explicit convention. The exact result, retaining the requested order, is ([[], [0, 1, 2, 3, 4]], [], Yes, exhaustive_complete_finite_topology)."},"skills":["Exhaust the Complete Finite Topology before Certifying Connectedness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r05-connected-topological-space-0068","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"023d02d13c53cdbc196dcf7f6025d34e636b3ba2d69be3d7484591a46adac530","family_id":"qf-mod34-connected-exhaustive","seed":202611035068,"semantic_template":"qf-mod34-connected-exhaustive/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,3,4],[0,3],[0]]}"},"fields":{"title":"Exhaust the Complete Finite Topology before Certifying Connectedness","stem":"Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 3], [0, 3, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Give (clopen_sets,nontrivial_clopen,connected?,exhaustive_complete_finite_topology), retaining the supplied topology order within both lists.","answer":{"kind":"expression","value":"([[], [0, 1, 2, 3, 4]], [], Yes, exhaustive_complete_finite_topology)","explanation":"Check the complement of every declared open subset and list every clopen set. Remove empty and whole sets to form the nontrivial list. Exactly an empty nontrivial list certifies connectedness under this explicit convention. The exact result, retaining the requested order, is ([[], [0, 1, 2, 3, 4]], [], Yes, exhaustive_complete_finite_topology)."},"skills":["Exhaust the Complete Finite Topology before Certifying Connectedness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r05-connected-topological-space-0069","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0f1f722488f72207a1544392ea4f500beef785e1e172c7e9e25703017c0419c8","family_id":"qf-mod34-connected-exhaustive","seed":202611035069,"semantic_template":"qf-mod34-connected-exhaustive/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,4],[0,4],[0]]}"},"fields":{"title":"Exhaust the Complete Finite Topology before Certifying Connectedness","stem":"Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 4], [0, 1, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Give (clopen_sets,nontrivial_clopen,connected?,exhaustive_complete_finite_topology), retaining the supplied topology order within both lists.","answer":{"kind":"expression","value":"([[], [0, 1, 2, 3, 4]], [], Yes, exhaustive_complete_finite_topology)","explanation":"Check the complement of every declared open subset and list every clopen set. Remove empty and whole sets to form the nontrivial list. Exactly an empty nontrivial list certifies connectedness under this explicit convention. The exact result, retaining the requested order, is ([[], [0, 1, 2, 3, 4]], [], Yes, exhaustive_complete_finite_topology)."},"skills":["Exhaust the Complete Finite Topology before Certifying Connectedness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r05-connected-topological-space-0070","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b97433bd1dd30fd6728a7ae5f8b4c61a8f597692a9fd66da2671ec17b5a14cc7","family_id":"qf-mod34-connected-exhaustive","seed":202611035070,"semantic_template":"qf-mod34-connected-exhaustive/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,4],[0,4],[0]]}"},"fields":{"title":"Exhaust the Complete Finite Topology before Certifying Connectedness","stem":"Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 4], [0, 1, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Give (clopen_sets,nontrivial_clopen,connected?,exhaustive_complete_finite_topology), retaining the supplied topology order within both lists.","answer":{"kind":"expression","value":"([[], [0, 1, 2, 3, 4]], [], Yes, exhaustive_complete_finite_topology)","explanation":"Check the complement of every declared open subset and list every clopen set. Remove empty and whole sets to form the nontrivial list. Exactly an empty nontrivial list certifies connectedness under this explicit convention. The exact result, retaining the requested order, is ([[], [0, 1, 2, 3, 4]], [], Yes, exhaustive_complete_finite_topology)."},"skills":["Exhaust the Complete Finite Topology before Certifying Connectedness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r05-connected-topological-space-0071","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bf480d494dde1c3f10f8f5ced98dbe00ee6a0a924b4a2451d720d3a8ab675874","family_id":"qf-mod34-connected-exhaustive","seed":202611035071,"semantic_template":"qf-mod34-connected-exhaustive/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,2,4],[0,4],[0]]}"},"fields":{"title":"Exhaust the Complete Finite Topology before Certifying Connectedness","stem":"Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 4], [0, 2, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Give (clopen_sets,nontrivial_clopen,connected?,exhaustive_complete_finite_topology), retaining the supplied topology order within both lists.","answer":{"kind":"expression","value":"([[], [0, 1, 2, 3, 4]], [], Yes, exhaustive_complete_finite_topology)","explanation":"Check the complement of every declared open subset and list every clopen set. Remove empty and whole sets to form the nontrivial list. Exactly an empty nontrivial list certifies connectedness under this explicit convention. The exact result, retaining the requested order, is ([[], [0, 1, 2, 3, 4]], [], Yes, exhaustive_complete_finite_topology)."},"skills":["Exhaust the Complete Finite Topology before Certifying Connectedness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r05-connected-topological-space-0072","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ae7be746d7bdacf3e4c16a7df7afc3f6e240206299aaf7aa07d76a0d05fc6c0a","family_id":"qf-mod34-connected-exhaustive","seed":202611035072,"semantic_template":"qf-mod34-connected-exhaustive/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,4],[0,4],[0]]}"},"fields":{"title":"Exhaust the Complete Finite Topology before Certifying Connectedness","stem":"Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 4], [0, 2, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Give (clopen_sets,nontrivial_clopen,connected?,exhaustive_complete_finite_topology), retaining the supplied topology order within both lists.","answer":{"kind":"expression","value":"([[], [0, 1, 2, 3, 4]], [], Yes, exhaustive_complete_finite_topology)","explanation":"Check the complement of every declared open subset and list every clopen set. Remove empty and whole sets to form the nontrivial list. Exactly an empty nontrivial list certifies connectedness under this explicit convention. The exact result, retaining the requested order, is ([[], [0, 1, 2, 3, 4]], [], Yes, exhaustive_complete_finite_topology)."},"skills":["Exhaust the Complete Finite Topology before Certifying Connectedness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r05-connected-topological-space-0073","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a0918dd28124ba1b295dadf560e95ab80c31d3deb6187781fe16192a8c5ad044","family_id":"qf-mod34-connected-exhaustive","seed":202611035073,"semantic_template":"qf-mod34-connected-exhaustive/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,3,4],[0,4],[0]]}"},"fields":{"title":"Exhaust the Complete Finite Topology before Certifying Connectedness","stem":"Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 4], [0, 3, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Give (clopen_sets,nontrivial_clopen,connected?,exhaustive_complete_finite_topology), retaining the supplied topology order within both lists.","answer":{"kind":"expression","value":"([[], [0, 1, 2, 3, 4]], [], Yes, exhaustive_complete_finite_topology)","explanation":"Check the complement of every declared open subset and list every clopen set. Remove empty and whole sets to form the nontrivial list. Exactly an empty nontrivial list certifies connectedness under this explicit convention. The exact result, retaining the requested order, is ([[], [0, 1, 2, 3, 4]], [], Yes, exhaustive_complete_finite_topology)."},"skills":["Exhaust the Complete Finite Topology before Certifying Connectedness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r05-connected-topological-space-0074","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4185e2fe1e5ef327d896e8c38a15f1866b0e9a9458088657949e036c08b1c1a6","family_id":"qf-mod34-connected-exhaustive","seed":202611035074,"semantic_template":"qf-mod34-connected-exhaustive/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,3,4],[0,4],[0]]}"},"fields":{"title":"Exhaust the Complete Finite Topology before Certifying Connectedness","stem":"Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 4], [0, 3, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Give (clopen_sets,nontrivial_clopen,connected?,exhaustive_complete_finite_topology), retaining the supplied topology order within both lists.","answer":{"kind":"expression","value":"([[], [0, 1, 2, 3, 4]], [], Yes, exhaustive_complete_finite_topology)","explanation":"Check the complement of every declared open subset and list every clopen set. Remove empty and whole sets to form the nontrivial list. Exactly an empty nontrivial list certifies connectedness under this explicit convention. The exact result, retaining the requested order, is ([[], [0, 1, 2, 3, 4]], [], Yes, exhaustive_complete_finite_topology)."},"skills":["Exhaust the Complete Finite Topology before Certifying Connectedness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r05-connected-topological-space-0075","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5d0d90fbec8e14d8c79d864c1c28a1b57aed0441acd159d6321b322935069101","family_id":"qf-mod34-connected-exhaustive","seed":202611035075,"semantic_template":"qf-mod34-connected-exhaustive/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2],[0,1],[1]]}"},"fields":{"title":"Exhaust the Complete Finite Topology before Certifying Connectedness","stem":"Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [1], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Give (clopen_sets,nontrivial_clopen,connected?,exhaustive_complete_finite_topology), retaining the supplied topology order within both lists.","answer":{"kind":"expression","value":"([[], [0, 1, 2, 3, 4]], [], Yes, exhaustive_complete_finite_topology)","explanation":"Check the complement of every declared open subset and list every clopen set. Remove empty and whole sets to form the nontrivial list. Exactly an empty nontrivial list certifies connectedness under this explicit convention. The exact result, retaining the requested order, is ([[], [0, 1, 2, 3, 4]], [], Yes, exhaustive_complete_finite_topology)."},"skills":["Exhaust the Complete Finite Topology before Certifying Connectedness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r05-connected-topological-space-0076","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"82b303b33a2f17cef26b1583a11c8348f0f5345fc87782c84d303fcb0b5b0ffe","family_id":"qf-mod34-connected-empty-singleton-boundary","seed":202611035076,"semantic_template":"qf-mod34-connected-empty-singleton-boundary/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2],[0,1],[0]],\"which\":\"empty\"}"},"fields":{"title":"Separate the Actual Finite Space from an Independent Reference Convention","stem":"Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Independent reference_kind=empty. This label selects only the empty-space or singleton reference convention; it does not say that the actual 5-point carrier equals that reference. Give (reference_kind,actual_carrier_size,actual_connected?,reference_convention). reference_convention is empty_space_connected_by_declared_vacuous_convention or singleton_connected according to the independent reference label.","answer":{"kind":"expression","value":"(empty, 5, Yes, empty_space_connected_by_declared_vacuous_convention)","explanation":"Compute actual_carrier_size and actual_connected only from the actual complete finite topology. Report the separate named reference convention without using it as a certificate for this actual space. Even a disconnected actual space can be compared with the true empty/singleton reference convention. The exact result, retaining the requested order, is (empty, 5, Yes, empty_space_connected_by_declared_vacuous_convention)."},"skills":["Separate the Actual Finite Space from an Independent Reference Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r05-connected-topological-space-0077","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d5bcb7d13392db271370d5a6e36aabef3b0f6c7f48b207d5cb4a27c2e2ba4385","family_id":"qf-mod34-connected-empty-singleton-boundary","seed":202611035077,"semantic_template":"qf-mod34-connected-empty-singleton-boundary/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,2],[0,1],[0]],\"which\":\"singleton\"}"},"fields":{"title":"Separate the Actual Finite Space from an Independent Reference Convention","stem":"Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Independent reference_kind=singleton. This label selects only the empty-space or singleton reference convention; it does not say that the actual 5-point carrier equals that reference. Give (reference_kind,actual_carrier_size,actual_connected?,reference_convention). reference_convention is empty_space_connected_by_declared_vacuous_convention or singleton_connected according to the independent reference label.","answer":{"kind":"expression","value":"(singleton, 5, Yes, singleton_connected)","explanation":"Compute actual_carrier_size and actual_connected only from the actual complete finite topology. Report the separate named reference convention without using it as a certificate for this actual space. Even a disconnected actual space can be compared with the true empty/singleton reference convention. The exact result, retaining the requested order, is (singleton, 5, Yes, singleton_connected)."},"skills":["Separate the Actual Finite Space from an Independent Reference Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r05-connected-topological-space-0078","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5baee800c9cc4bd2b4cd6c328281c2f01d80f04a0e646cd6269906223266cac9","family_id":"qf-mod34-connected-empty-singleton-boundary","seed":202611035078,"semantic_template":"qf-mod34-connected-empty-singleton-boundary/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,3],[0,1],[0]],\"which\":\"empty\"}"},"fields":{"title":"Separate the Actual Finite Space from an Independent Reference Convention","stem":"Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 1], [0, 1, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Independent reference_kind=empty. This label selects only the empty-space or singleton reference convention; it does not say that the actual 5-point carrier equals that reference. Give (reference_kind,actual_carrier_size,actual_connected?,reference_convention). reference_convention is empty_space_connected_by_declared_vacuous_convention or singleton_connected according to the independent reference label.","answer":{"kind":"expression","value":"(empty, 5, Yes, empty_space_connected_by_declared_vacuous_convention)","explanation":"Compute actual_carrier_size and actual_connected only from the actual complete finite topology. Report the separate named reference convention without using it as a certificate for this actual space. Even a disconnected actual space can be compared with the true empty/singleton reference convention. The exact result, retaining the requested order, is (empty, 5, Yes, empty_space_connected_by_declared_vacuous_convention)."},"skills":["Separate the Actual Finite Space from an Independent Reference Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r05-connected-topological-space-0079","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7030731bd0558c0319a7f32c3b78200b9661a147ac218c025ea2ed6de103f244","family_id":"qf-mod34-connected-empty-singleton-boundary","seed":202611035079,"semantic_template":"qf-mod34-connected-empty-singleton-boundary/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,3],[0,1],[0]],\"which\":\"singleton\"}"},"fields":{"title":"Separate the Actual Finite Space from an Independent Reference Convention","stem":"Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 1], [0, 1, 3], [0, 1, 3, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Independent reference_kind=singleton. This label selects only the empty-space or singleton reference convention; it does not say that the actual 5-point carrier equals that reference. Give (reference_kind,actual_carrier_size,actual_connected?,reference_convention). reference_convention is empty_space_connected_by_declared_vacuous_convention or singleton_connected according to the independent reference label.","answer":{"kind":"expression","value":"(singleton, 5, Yes, singleton_connected)","explanation":"Compute actual_carrier_size and actual_connected only from the actual complete finite topology. Report the separate named reference convention without using it as a certificate for this actual space. Even a disconnected actual space can be compared with the true empty/singleton reference convention. The exact result, retaining the requested order, is (singleton, 5, Yes, singleton_connected)."},"skills":["Separate the Actual Finite Space from an Independent Reference Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r05-connected-topological-space-0080","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"78068832101b954096f01a8be124cb594b539c7890179d50176816588f046aca","family_id":"qf-mod34-connected-empty-singleton-boundary","seed":202611035080,"semantic_template":"qf-mod34-connected-empty-singleton-boundary/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,4],[0,1],[0]],\"which\":\"empty\"}"},"fields":{"title":"Separate the Actual Finite Space from an Independent Reference Convention","stem":"Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 1], [0, 1, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Independent reference_kind=empty. This label selects only the empty-space or singleton reference convention; it does not say that the actual 5-point carrier equals that reference. Give (reference_kind,actual_carrier_size,actual_connected?,reference_convention). reference_convention is empty_space_connected_by_declared_vacuous_convention or singleton_connected according to the independent reference label.","answer":{"kind":"expression","value":"(empty, 5, Yes, empty_space_connected_by_declared_vacuous_convention)","explanation":"Compute actual_carrier_size and actual_connected only from the actual complete finite topology. Report the separate named reference convention without using it as a certificate for this actual space. Even a disconnected actual space can be compared with the true empty/singleton reference convention. The exact result, retaining the requested order, is (empty, 5, Yes, empty_space_connected_by_declared_vacuous_convention)."},"skills":["Separate the Actual Finite Space from an Independent Reference Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r05-connected-topological-space-0081","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a80179fe027d2e15ff763c2e9918a160e6d4c881ab73ef3f1813cce6b125b93d","family_id":"qf-mod34-connected-empty-singleton-boundary","seed":202611035081,"semantic_template":"qf-mod34-connected-empty-singleton-boundary/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,4],[0,1],[0]],\"which\":\"singleton\"}"},"fields":{"title":"Separate the Actual Finite Space from an Independent Reference Convention","stem":"Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 1], [0, 1, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Independent reference_kind=singleton. This label selects only the empty-space or singleton reference convention; it does not say that the actual 5-point carrier equals that reference. Give (reference_kind,actual_carrier_size,actual_connected?,reference_convention). reference_convention is empty_space_connected_by_declared_vacuous_convention or singleton_connected according to the independent reference label.","answer":{"kind":"expression","value":"(singleton, 5, Yes, singleton_connected)","explanation":"Compute actual_carrier_size and actual_connected only from the actual complete finite topology. Report the separate named reference convention without using it as a certificate for this actual space. Even a disconnected actual space can be compared with the true empty/singleton reference convention. The exact result, retaining the requested order, is (singleton, 5, Yes, singleton_connected)."},"skills":["Separate the Actual Finite Space from an Independent Reference Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r05-connected-topological-space-0082","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b4dc41d428d839d244a8495ada614eb7fc5302d477ea6acc0ece22fb577c95c9","family_id":"qf-mod34-connected-empty-singleton-boundary","seed":202611035082,"semantic_template":"qf-mod34-connected-empty-singleton-boundary/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2],[0,2],[0]],\"which\":\"empty\"}"},"fields":{"title":"Separate the Actual Finite Space from an Independent Reference Convention","stem":"Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 2], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Independent reference_kind=empty. This label selects only the empty-space or singleton reference convention; it does not say that the actual 5-point carrier equals that reference. Give (reference_kind,actual_carrier_size,actual_connected?,reference_convention). reference_convention is empty_space_connected_by_declared_vacuous_convention or singleton_connected according to the independent reference label.","answer":{"kind":"expression","value":"(empty, 5, Yes, empty_space_connected_by_declared_vacuous_convention)","explanation":"Compute actual_carrier_size and actual_connected only from the actual complete finite topology. Report the separate named reference convention without using it as a certificate for this actual space. Even a disconnected actual space can be compared with the true empty/singleton reference convention. The exact result, retaining the requested order, is (empty, 5, Yes, empty_space_connected_by_declared_vacuous_convention)."},"skills":["Separate the Actual Finite Space from an Independent Reference Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r05-connected-topological-space-0083","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2b49a34ff52d0aefbc14e39b8b75b4ca7352601c307994b1af5d04937d36141c","family_id":"qf-mod34-connected-empty-singleton-boundary","seed":202611035083,"semantic_template":"qf-mod34-connected-empty-singleton-boundary/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,2],[0,2],[0]],\"which\":\"singleton\"}"},"fields":{"title":"Separate the Actual Finite Space from an Independent Reference Convention","stem":"Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 2], [0, 1, 2], [0, 1, 2, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Independent reference_kind=singleton. This label selects only the empty-space or singleton reference convention; it does not say that the actual 5-point carrier equals that reference. Give (reference_kind,actual_carrier_size,actual_connected?,reference_convention). reference_convention is empty_space_connected_by_declared_vacuous_convention or singleton_connected according to the independent reference label.","answer":{"kind":"expression","value":"(singleton, 5, Yes, singleton_connected)","explanation":"Compute actual_carrier_size and actual_connected only from the actual complete finite topology. Report the separate named reference convention without using it as a certificate for this actual space. Even a disconnected actual space can be compared with the true empty/singleton reference convention. The exact result, retaining the requested order, is (singleton, 5, Yes, singleton_connected)."},"skills":["Separate the Actual Finite Space from an Independent Reference Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r05-connected-topological-space-0084","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ff0b3ed991f11855ad23c0ea2a1dfcfbf03cd5c87a59653df51c62f1f4f0bc38","family_id":"qf-mod34-connected-empty-singleton-boundary","seed":202611035084,"semantic_template":"qf-mod34-connected-empty-singleton-boundary/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,2,3],[0,2],[0]],\"which\":\"empty\"}"},"fields":{"title":"Separate the Actual Finite Space from an Independent Reference Convention","stem":"Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 2], [0, 2, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Independent reference_kind=empty. This label selects only the empty-space or singleton reference convention; it does not say that the actual 5-point carrier equals that reference. Give (reference_kind,actual_carrier_size,actual_connected?,reference_convention). reference_convention is empty_space_connected_by_declared_vacuous_convention or singleton_connected according to the independent reference label.","answer":{"kind":"expression","value":"(empty, 5, Yes, empty_space_connected_by_declared_vacuous_convention)","explanation":"Compute actual_carrier_size and actual_connected only from the actual complete finite topology. Report the separate named reference convention without using it as a certificate for this actual space. Even a disconnected actual space can be compared with the true empty/singleton reference convention. The exact result, retaining the requested order, is (empty, 5, Yes, empty_space_connected_by_declared_vacuous_convention)."},"skills":["Separate the Actual Finite Space from an Independent Reference Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r05-connected-topological-space-0085","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0e917c83edae31566adc6ca07425878cb0b0028c62c20d3fae7fa05733f1136d","family_id":"qf-mod34-connected-empty-singleton-boundary","seed":202611035085,"semantic_template":"qf-mod34-connected-empty-singleton-boundary/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,3],[0,2],[0]],\"which\":\"singleton\"}"},"fields":{"title":"Separate the Actual Finite Space from an Independent Reference Convention","stem":"Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Independent reference_kind=singleton. This label selects only the empty-space or singleton reference convention; it does not say that the actual 5-point carrier equals that reference. Give (reference_kind,actual_carrier_size,actual_connected?,reference_convention). reference_convention is empty_space_connected_by_declared_vacuous_convention or singleton_connected according to the independent reference label.","answer":{"kind":"expression","value":"(singleton, 5, Yes, singleton_connected)","explanation":"Compute actual_carrier_size and actual_connected only from the actual complete finite topology. Report the separate named reference convention without using it as a certificate for this actual space. Even a disconnected actual space can be compared with the true empty/singleton reference convention. The exact result, retaining the requested order, is (singleton, 5, Yes, singleton_connected)."},"skills":["Separate the Actual Finite Space from an Independent Reference Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r05-connected-topological-space-0086","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bcc728eaf08f0856da14fd3c6ca6b5cb51e4b1b117426c44a721e3f968e517f1","family_id":"qf-mod34-connected-empty-singleton-boundary","seed":202611035086,"semantic_template":"qf-mod34-connected-empty-singleton-boundary/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,2,4],[0,2],[0]],\"which\":\"empty\"}"},"fields":{"title":"Separate the Actual Finite Space from an Independent Reference Convention","stem":"Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 2], [0, 2, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Independent reference_kind=empty. This label selects only the empty-space or singleton reference convention; it does not say that the actual 5-point carrier equals that reference. Give (reference_kind,actual_carrier_size,actual_connected?,reference_convention). reference_convention is empty_space_connected_by_declared_vacuous_convention or singleton_connected according to the independent reference label.","answer":{"kind":"expression","value":"(empty, 5, Yes, empty_space_connected_by_declared_vacuous_convention)","explanation":"Compute actual_carrier_size and actual_connected only from the actual complete finite topology. Report the separate named reference convention without using it as a certificate for this actual space. Even a disconnected actual space can be compared with the true empty/singleton reference convention. The exact result, retaining the requested order, is (empty, 5, Yes, empty_space_connected_by_declared_vacuous_convention)."},"skills":["Separate the Actual Finite Space from an Independent Reference Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r05-connected-topological-space-0087","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e9db191b4ae0ca89b95ac6bdd9fe366a8f0d6473f194b165ebd6dce0504faa0e","family_id":"qf-mod34-connected-empty-singleton-boundary","seed":202611035087,"semantic_template":"qf-mod34-connected-empty-singleton-boundary/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,4],[0,2],[0]],\"which\":\"singleton\"}"},"fields":{"title":"Separate the Actual Finite Space from an Independent Reference Convention","stem":"Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 2], [0, 2, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Independent reference_kind=singleton. This label selects only the empty-space or singleton reference convention; it does not say that the actual 5-point carrier equals that reference. Give (reference_kind,actual_carrier_size,actual_connected?,reference_convention). reference_convention is empty_space_connected_by_declared_vacuous_convention or singleton_connected according to the independent reference label.","answer":{"kind":"expression","value":"(singleton, 5, Yes, singleton_connected)","explanation":"Compute actual_carrier_size and actual_connected only from the actual complete finite topology. Report the separate named reference convention without using it as a certificate for this actual space. Even a disconnected actual space can be compared with the true empty/singleton reference convention. The exact result, retaining the requested order, is (singleton, 5, Yes, singleton_connected)."},"skills":["Separate the Actual Finite Space from an Independent Reference Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r05-connected-topological-space-0088","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c833f95445fa76aa9bb5f2a088caf72e3766a19e384668f785e46e319623bd5c","family_id":"qf-mod34-connected-empty-singleton-boundary","seed":202611035088,"semantic_template":"qf-mod34-connected-empty-singleton-boundary/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,3],[0,3],[0]],\"which\":\"empty\"}"},"fields":{"title":"Separate the Actual Finite Space from an Independent Reference Convention","stem":"Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 3], [0, 1, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Independent reference_kind=empty. This label selects only the empty-space or singleton reference convention; it does not say that the actual 5-point carrier equals that reference. Give (reference_kind,actual_carrier_size,actual_connected?,reference_convention). reference_convention is empty_space_connected_by_declared_vacuous_convention or singleton_connected according to the independent reference label.","answer":{"kind":"expression","value":"(empty, 5, Yes, empty_space_connected_by_declared_vacuous_convention)","explanation":"Compute actual_carrier_size and actual_connected only from the actual complete finite topology. Report the separate named reference convention without using it as a certificate for this actual space. Even a disconnected actual space can be compared with the true empty/singleton reference convention. The exact result, retaining the requested order, is (empty, 5, Yes, empty_space_connected_by_declared_vacuous_convention)."},"skills":["Separate the Actual Finite Space from an Independent Reference Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r05-connected-topological-space-0089","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"65538a08874937fb2d2f84a05352d508eb52487d8cc4e77bdff7e001e2653748","family_id":"qf-mod34-connected-empty-singleton-boundary","seed":202611035089,"semantic_template":"qf-mod34-connected-empty-singleton-boundary/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,3],[0,3],[0]],\"which\":\"singleton\"}"},"fields":{"title":"Separate the Actual Finite Space from an Independent Reference Convention","stem":"Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 3], [0, 1, 3], [0, 1, 3, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Independent reference_kind=singleton. This label selects only the empty-space or singleton reference convention; it does not say that the actual 5-point carrier equals that reference. Give (reference_kind,actual_carrier_size,actual_connected?,reference_convention). reference_convention is empty_space_connected_by_declared_vacuous_convention or singleton_connected according to the independent reference label.","answer":{"kind":"expression","value":"(singleton, 5, Yes, singleton_connected)","explanation":"Compute actual_carrier_size and actual_connected only from the actual complete finite topology. Report the separate named reference convention without using it as a certificate for this actual space. Even a disconnected actual space can be compared with the true empty/singleton reference convention. The exact result, retaining the requested order, is (singleton, 5, Yes, singleton_connected)."},"skills":["Separate the Actual Finite Space from an Independent Reference Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r05-connected-topological-space-0090","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"df2bbf5be37bef6beb7ca0acf747a6333a67cb230ad98fece75b971aed2212a7","family_id":"qf-mod34-connected-empty-singleton-boundary","seed":202611035090,"semantic_template":"qf-mod34-connected-empty-singleton-boundary/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,2,3],[0,3],[0]],\"which\":\"empty\"}"},"fields":{"title":"Separate the Actual Finite Space from an Independent Reference Convention","stem":"Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 3], [0, 2, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Independent reference_kind=empty. This label selects only the empty-space or singleton reference convention; it does not say that the actual 5-point carrier equals that reference. Give (reference_kind,actual_carrier_size,actual_connected?,reference_convention). reference_convention is empty_space_connected_by_declared_vacuous_convention or singleton_connected according to the independent reference label.","answer":{"kind":"expression","value":"(empty, 5, Yes, empty_space_connected_by_declared_vacuous_convention)","explanation":"Compute actual_carrier_size and actual_connected only from the actual complete finite topology. Report the separate named reference convention without using it as a certificate for this actual space. Even a disconnected actual space can be compared with the true empty/singleton reference convention. The exact result, retaining the requested order, is (empty, 5, Yes, empty_space_connected_by_declared_vacuous_convention)."},"skills":["Separate the Actual Finite Space from an Independent Reference Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r05-connected-topological-space-0091","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f03af1b52b1c38f5b3a87bfd618f74084082813313ca47b370fb1bfa36abd628","family_id":"qf-mod34-connected-empty-singleton-boundary","seed":202611035091,"semantic_template":"qf-mod34-connected-empty-singleton-boundary/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,3],[0,3],[0]],\"which\":\"singleton\"}"},"fields":{"title":"Separate the Actual Finite Space from an Independent Reference Convention","stem":"Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 3], [0, 2, 3], [0, 2, 3, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Independent reference_kind=singleton. This label selects only the empty-space or singleton reference convention; it does not say that the actual 5-point carrier equals that reference. Give (reference_kind,actual_carrier_size,actual_connected?,reference_convention). reference_convention is empty_space_connected_by_declared_vacuous_convention or singleton_connected according to the independent reference label.","answer":{"kind":"expression","value":"(singleton, 5, Yes, singleton_connected)","explanation":"Compute actual_carrier_size and actual_connected only from the actual complete finite topology. Report the separate named reference convention without using it as a certificate for this actual space. Even a disconnected actual space can be compared with the true empty/singleton reference convention. The exact result, retaining the requested order, is (singleton, 5, Yes, singleton_connected)."},"skills":["Separate the Actual Finite Space from an Independent Reference Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r05-connected-topological-space-0092","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"04fd3fd7b40fa62ecb6d122bc35ab8598f3568c9ada6f352f345f73bf89b07a1","family_id":"qf-mod34-connected-empty-singleton-boundary","seed":202611035092,"semantic_template":"qf-mod34-connected-empty-singleton-boundary/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,3,4],[0,3],[0]],\"which\":\"empty\"}"},"fields":{"title":"Separate the Actual Finite Space from an Independent Reference Convention","stem":"Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 3], [0, 3, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Independent reference_kind=empty. This label selects only the empty-space or singleton reference convention; it does not say that the actual 5-point carrier equals that reference. Give (reference_kind,actual_carrier_size,actual_connected?,reference_convention). reference_convention is empty_space_connected_by_declared_vacuous_convention or singleton_connected according to the independent reference label.","answer":{"kind":"expression","value":"(empty, 5, Yes, empty_space_connected_by_declared_vacuous_convention)","explanation":"Compute actual_carrier_size and actual_connected only from the actual complete finite topology. Report the separate named reference convention without using it as a certificate for this actual space. Even a disconnected actual space can be compared with the true empty/singleton reference convention. The exact result, retaining the requested order, is (empty, 5, Yes, empty_space_connected_by_declared_vacuous_convention)."},"skills":["Separate the Actual Finite Space from an Independent Reference Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r05-connected-topological-space-0093","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d464364a22dd7b95b21d22b360cf3ca55327dccd48141f9b001e636bb2b60dd8","family_id":"qf-mod34-connected-empty-singleton-boundary","seed":202611035093,"semantic_template":"qf-mod34-connected-empty-singleton-boundary/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,3,4],[0,3],[0]],\"which\":\"singleton\"}"},"fields":{"title":"Separate the Actual Finite Space from an Independent Reference Convention","stem":"Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 3], [0, 3, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Independent reference_kind=singleton. This label selects only the empty-space or singleton reference convention; it does not say that the actual 5-point carrier equals that reference. Give (reference_kind,actual_carrier_size,actual_connected?,reference_convention). reference_convention is empty_space_connected_by_declared_vacuous_convention or singleton_connected according to the independent reference label.","answer":{"kind":"expression","value":"(singleton, 5, Yes, singleton_connected)","explanation":"Compute actual_carrier_size and actual_connected only from the actual complete finite topology. Report the separate named reference convention without using it as a certificate for this actual space. Even a disconnected actual space can be compared with the true empty/singleton reference convention. The exact result, retaining the requested order, is (singleton, 5, Yes, singleton_connected)."},"skills":["Separate the Actual Finite Space from an Independent Reference Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r05-connected-topological-space-0094","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9207373a0e5c151de1c783801f836eabf5664dbc7f483b71d9e3475f9428fabf","family_id":"qf-mod34-connected-empty-singleton-boundary","seed":202611035094,"semantic_template":"qf-mod34-connected-empty-singleton-boundary/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,4],[0,4],[0]],\"which\":\"empty\"}"},"fields":{"title":"Separate the Actual Finite Space from an Independent Reference Convention","stem":"Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 4], [0, 1, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Independent reference_kind=empty. This label selects only the empty-space or singleton reference convention; it does not say that the actual 5-point carrier equals that reference. Give (reference_kind,actual_carrier_size,actual_connected?,reference_convention). reference_convention is empty_space_connected_by_declared_vacuous_convention or singleton_connected according to the independent reference label.","answer":{"kind":"expression","value":"(empty, 5, Yes, empty_space_connected_by_declared_vacuous_convention)","explanation":"Compute actual_carrier_size and actual_connected only from the actual complete finite topology. Report the separate named reference convention without using it as a certificate for this actual space. Even a disconnected actual space can be compared with the true empty/singleton reference convention. The exact result, retaining the requested order, is (empty, 5, Yes, empty_space_connected_by_declared_vacuous_convention)."},"skills":["Separate the Actual Finite Space from an Independent Reference Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r05-connected-topological-space-0095","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bbc5cb2063a1b534acc2451150b7c7d410aa27968b285cf826dd8ddc7795737b","family_id":"qf-mod34-connected-empty-singleton-boundary","seed":202611035095,"semantic_template":"qf-mod34-connected-empty-singleton-boundary/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,4],[0,4],[0]],\"which\":\"singleton\"}"},"fields":{"title":"Separate the Actual Finite Space from an Independent Reference Convention","stem":"Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 4], [0, 1, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Independent reference_kind=singleton. This label selects only the empty-space or singleton reference convention; it does not say that the actual 5-point carrier equals that reference. Give (reference_kind,actual_carrier_size,actual_connected?,reference_convention). reference_convention is empty_space_connected_by_declared_vacuous_convention or singleton_connected according to the independent reference label.","answer":{"kind":"expression","value":"(singleton, 5, Yes, singleton_connected)","explanation":"Compute actual_carrier_size and actual_connected only from the actual complete finite topology. Report the separate named reference convention without using it as a certificate for this actual space. Even a disconnected actual space can be compared with the true empty/singleton reference convention. The exact result, retaining the requested order, is (singleton, 5, Yes, singleton_connected)."},"skills":["Separate the Actual Finite Space from an Independent Reference Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r05-connected-topological-space-0096","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7ba95267efe540d22129a2d4a01fd0edc132dff318fa5bebf19a301259827baf","family_id":"qf-mod34-connected-empty-singleton-boundary","seed":202611035096,"semantic_template":"qf-mod34-connected-empty-singleton-boundary/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,2,4],[0,4],[0]],\"which\":\"empty\"}"},"fields":{"title":"Separate the Actual Finite Space from an Independent Reference Convention","stem":"Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 4], [0, 2, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Independent reference_kind=empty. This label selects only the empty-space or singleton reference convention; it does not say that the actual 5-point carrier equals that reference. Give (reference_kind,actual_carrier_size,actual_connected?,reference_convention). reference_convention is empty_space_connected_by_declared_vacuous_convention or singleton_connected according to the independent reference label.","answer":{"kind":"expression","value":"(empty, 5, Yes, empty_space_connected_by_declared_vacuous_convention)","explanation":"Compute actual_carrier_size and actual_connected only from the actual complete finite topology. Report the separate named reference convention without using it as a certificate for this actual space. Even a disconnected actual space can be compared with the true empty/singleton reference convention. The exact result, retaining the requested order, is (empty, 5, Yes, empty_space_connected_by_declared_vacuous_convention)."},"skills":["Separate the Actual Finite Space from an Independent Reference Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r05-connected-topological-space-0097","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4cc29b3bd083b1edbf8b7212fcc5beb44d30bff54373dd6d62e97247af5ca171","family_id":"qf-mod34-connected-empty-singleton-boundary","seed":202611035097,"semantic_template":"qf-mod34-connected-empty-singleton-boundary/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,4],[0,4],[0]],\"which\":\"singleton\"}"},"fields":{"title":"Separate the Actual Finite Space from an Independent Reference Convention","stem":"Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 4], [0, 2, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Independent reference_kind=singleton. This label selects only the empty-space or singleton reference convention; it does not say that the actual 5-point carrier equals that reference. Give (reference_kind,actual_carrier_size,actual_connected?,reference_convention). reference_convention is empty_space_connected_by_declared_vacuous_convention or singleton_connected according to the independent reference label.","answer":{"kind":"expression","value":"(singleton, 5, Yes, singleton_connected)","explanation":"Compute actual_carrier_size and actual_connected only from the actual complete finite topology. Report the separate named reference convention without using it as a certificate for this actual space. Even a disconnected actual space can be compared with the true empty/singleton reference convention. The exact result, retaining the requested order, is (singleton, 5, Yes, singleton_connected)."},"skills":["Separate the Actual Finite Space from an Independent Reference Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r05-connected-topological-space-0098","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"54467b1518be6b2d44ed8879e3bb07fe2c6f0c8393daa567aac8283d8da7f3f9","family_id":"qf-mod34-connected-empty-singleton-boundary","seed":202611035098,"semantic_template":"qf-mod34-connected-empty-singleton-boundary/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,3,4],[0,4],[0]],\"which\":\"empty\"}"},"fields":{"title":"Separate the Actual Finite Space from an Independent Reference Convention","stem":"Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 4], [0, 3, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Independent reference_kind=empty. This label selects only the empty-space or singleton reference convention; it does not say that the actual 5-point carrier equals that reference. Give (reference_kind,actual_carrier_size,actual_connected?,reference_convention). reference_convention is empty_space_connected_by_declared_vacuous_convention or singleton_connected according to the independent reference label.","answer":{"kind":"expression","value":"(empty, 5, Yes, empty_space_connected_by_declared_vacuous_convention)","explanation":"Compute actual_carrier_size and actual_connected only from the actual complete finite topology. Report the separate named reference convention without using it as a certificate for this actual space. Even a disconnected actual space can be compared with the true empty/singleton reference convention. The exact result, retaining the requested order, is (empty, 5, Yes, empty_space_connected_by_declared_vacuous_convention)."},"skills":["Separate the Actual Finite Space from an Independent Reference Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r05-connected-topological-space-0099","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"719ba8e15690b43c7e845f4e79fb02d345a422d385644009f1253814a558f8ba","family_id":"qf-mod34-connected-empty-singleton-boundary","seed":202611035099,"semantic_template":"qf-mod34-connected-empty-singleton-boundary/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,3,4],[0,4],[0]],\"which\":\"singleton\"}"},"fields":{"title":"Separate the Actual Finite Space from an Independent Reference Convention","stem":"Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 4], [0, 3, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Independent reference_kind=singleton. This label selects only the empty-space or singleton reference convention; it does not say that the actual 5-point carrier equals that reference. Give (reference_kind,actual_carrier_size,actual_connected?,reference_convention). reference_convention is empty_space_connected_by_declared_vacuous_convention or singleton_connected according to the independent reference label.","answer":{"kind":"expression","value":"(singleton, 5, Yes, singleton_connected)","explanation":"Compute actual_carrier_size and actual_connected only from the actual complete finite topology. Report the separate named reference convention without using it as a certificate for this actual space. Even a disconnected actual space can be compared with the true empty/singleton reference convention. The exact result, retaining the requested order, is (singleton, 5, Yes, singleton_connected)."},"skills":["Separate the Actual Finite Space from an Independent Reference Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r05-connected-topological-space-0100","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"476cf47957ad79f89e0a25452b04b3a501a51cb46bf19077899805e081681b75","family_id":"qf-mod34-connected-empty-singleton-boundary","seed":202611035100,"semantic_template":"qf-mod34-connected-empty-singleton-boundary/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2],[0,1],[1]],\"which\":\"empty\"}"},"fields":{"title":"Separate the Actual Finite Space from an Independent Reference Convention","stem":"Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [1], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Independent reference_kind=empty. This label selects only the empty-space or singleton reference convention; it does not say that the actual 5-point carrier equals that reference. Give (reference_kind,actual_carrier_size,actual_connected?,reference_convention). reference_convention is empty_space_connected_by_declared_vacuous_convention or singleton_connected according to the independent reference label.","answer":{"kind":"expression","value":"(empty, 5, Yes, empty_space_connected_by_declared_vacuous_convention)","explanation":"Compute actual_carrier_size and actual_connected only from the actual complete finite topology. Report the separate named reference convention without using it as a certificate for this actual space. Even a disconnected actual space can be compared with the true empty/singleton reference convention. The exact result, retaining the requested order, is (empty, 5, Yes, empty_space_connected_by_declared_vacuous_convention)."},"skills":["Separate the Actual Finite Space from an Independent Reference Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}