{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r03-subspace-topology-0001","batch_id":"batch-20261014-advance-20261108-r03-subspace-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8e5395ceefc270320a18038f6dc3d5592dc28a814a83a2f6db5a112c6423816e","family_id":"qf-mod39-subtop-induced-topology","seed":202611083001,"semantic_template":"qf-mod39-subtop-induced-topology/{\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2],[0,1],[0]],\"subset\":[1,3,4]}"},"fields":{"title":"Normalize All Ambient-Open Intersections to Obtain the Relative Topology","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]], and subset Y=[1, 3, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Give (subset, induced_topology, open_count, topology_certificate, certificate).","answer":{"kind":"expression","value":"([1, 3, 4], [[], [1], [1, 3], [1, 3, 4]], 4, Yes, all_parent_open_intersections_normalized)","explanation":"Intersect every parent open with Y, identify duplicate intersections and retain the source canonical family order. The empty subspace has the one-open-set topology {empty set}. This finite complete construction does not sample an unspecified ambient topology. The exact result, retaining the requested order, is ([1, 3, 4], [[], [1], [1, 3], [1, 3, 4]], 4, Yes, all_parent_open_intersections_normalized)."},"skills":["Normalize All Ambient-Open Intersections to Obtain the Relative Topology"],"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-20261108-r03-subspace-topology-0002","batch_id":"batch-20261014-advance-20261108-r03-subspace-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"177f46ac9ffed1920db977ec85180e9a4be96508af3491c97a59de7c9d2daa13","family_id":"qf-mod39-subtop-induced-topology","seed":202611083002,"semantic_template":"qf-mod39-subtop-induced-topology/{\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,2],[0,1],[0]],\"subset\":[1,3,4]}"},"fields":{"title":"Normalize All Ambient-Open Intersections to Obtain the Relative Topology","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3, 4], [0, 1, 2, 4]], and subset Y=[1, 3, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Give (subset, induced_topology, open_count, topology_certificate, certificate).","answer":{"kind":"expression","value":"([1, 3, 4], [[], [1], [1, 3, 4], [1, 4]], 4, Yes, all_parent_open_intersections_normalized)","explanation":"Intersect every parent open with Y, identify duplicate intersections and retain the source canonical family order. The empty subspace has the one-open-set topology {empty set}. This finite complete construction does not sample an unspecified ambient topology. The exact result, retaining the requested order, is ([1, 3, 4], [[], [1], [1, 3, 4], [1, 4]], 4, Yes, all_parent_open_intersections_normalized)."},"skills":["Normalize All Ambient-Open Intersections to Obtain the Relative Topology"],"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-20261108-r03-subspace-topology-0003","batch_id":"batch-20261014-advance-20261108-r03-subspace-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"132c0c4443d3f51feb22a8c0cbb78ca582742c1f38794167aaafa84954db8480","family_id":"qf-mod39-subtop-induced-topology","seed":202611083003,"semantic_template":"qf-mod39-subtop-induced-topology/{\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,3],[0,1],[0]],\"subset\":[1,2,4]}"},"fields":{"title":"Normalize All Ambient-Open Intersections to Obtain the Relative Topology","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 3]], and subset Y=[1, 2, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Give (subset, induced_topology, open_count, topology_certificate, certificate).","answer":{"kind":"expression","value":"([1, 2, 4], [[], [1], [1, 2], [1, 2, 4]], 4, Yes, all_parent_open_intersections_normalized)","explanation":"Intersect every parent open with Y, identify duplicate intersections and retain the source canonical family order. The empty subspace has the one-open-set topology {empty set}. This finite complete construction does not sample an unspecified ambient topology. The exact result, retaining the requested order, is ([1, 2, 4], [[], [1], [1, 2], [1, 2, 4]], 4, Yes, all_parent_open_intersections_normalized)."},"skills":["Normalize All Ambient-Open Intersections to Obtain the Relative Topology"],"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-20261108-r03-subspace-topology-0004","batch_id":"batch-20261014-advance-20261108-r03-subspace-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"40f1729de097ffb84fd58938fa26be9282871d931b5ca67d0114d49ccd085eb0","family_id":"qf-mod39-subtop-induced-topology","seed":202611083004,"semantic_template":"qf-mod39-subtop-induced-topology/{\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,3],[0,1],[0]],\"subset\":[1,2,4]}"},"fields":{"title":"Normalize All Ambient-Open Intersections to Obtain the Relative Topology","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1], [0, 1, 2, 3, 4], [0, 1, 3], [0, 1, 3, 4]], and subset Y=[1, 2, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Give (subset, induced_topology, open_count, topology_certificate, certificate).","answer":{"kind":"expression","value":"([1, 2, 4], [[], [1], [1, 2, 4], [1, 4]], 4, Yes, all_parent_open_intersections_normalized)","explanation":"Intersect every parent open with Y, identify duplicate intersections and retain the source canonical family order. The empty subspace has the one-open-set topology {empty set}. This finite complete construction does not sample an unspecified ambient topology. The exact result, retaining the requested order, is ([1, 2, 4], [[], [1], [1, 2, 4], [1, 4]], 4, Yes, all_parent_open_intersections_normalized)."},"skills":["Normalize All Ambient-Open Intersections to Obtain the Relative Topology"],"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-20261108-r03-subspace-topology-0005","batch_id":"batch-20261014-advance-20261108-r03-subspace-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dcf454703c2ac09be0eaddca83940df54de54c2e4f81952bacea051912b1a137","family_id":"qf-mod39-subtop-induced-topology","seed":202611083005,"semantic_template":"qf-mod39-subtop-induced-topology/{\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,4],[0,1],[0]],\"subset\":[1,2,3]}"},"fields":{"title":"Normalize All Ambient-Open Intersections to Obtain the Relative Topology","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 4]], and subset Y=[1, 2, 3]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Give (subset, induced_topology, open_count, topology_certificate, certificate).","answer":{"kind":"expression","value":"([1, 2, 3], [[], [1], [1, 2], [1, 2, 3]], 4, Yes, all_parent_open_intersections_normalized)","explanation":"Intersect every parent open with Y, identify duplicate intersections and retain the source canonical family order. The empty subspace has the one-open-set topology {empty set}. This finite complete construction does not sample an unspecified ambient topology. The exact result, retaining the requested order, is ([1, 2, 3], [[], [1], [1, 2], [1, 2, 3]], 4, Yes, all_parent_open_intersections_normalized)."},"skills":["Normalize All Ambient-Open Intersections to Obtain the Relative Topology"],"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-20261108-r03-subspace-topology-0006","batch_id":"batch-20261014-advance-20261108-r03-subspace-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3ff1c2f820defe257942161437ee22ed509d88cf60761be3334b367a7aa3c235","family_id":"qf-mod39-subtop-induced-topology","seed":202611083006,"semantic_template":"qf-mod39-subtop-induced-topology/{\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,4],[0,1],[0]],\"subset\":[1,2,3]}"},"fields":{"title":"Normalize All Ambient-Open Intersections to Obtain the Relative Topology","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1], [0, 1, 2, 3, 4], [0, 1, 3, 4], [0, 1, 4]], and subset Y=[1, 2, 3]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Give (subset, induced_topology, open_count, topology_certificate, certificate).","answer":{"kind":"expression","value":"([1, 2, 3], [[], [1], [1, 2, 3], [1, 3]], 4, Yes, all_parent_open_intersections_normalized)","explanation":"Intersect every parent open with Y, identify duplicate intersections and retain the source canonical family order. The empty subspace has the one-open-set topology {empty set}. This finite complete construction does not sample an unspecified ambient topology. The exact result, retaining the requested order, is ([1, 2, 3], [[], [1], [1, 2, 3], [1, 3]], 4, Yes, all_parent_open_intersections_normalized)."},"skills":["Normalize All Ambient-Open Intersections to Obtain the Relative Topology"],"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-20261108-r03-subspace-topology-0007","batch_id":"batch-20261014-advance-20261108-r03-subspace-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d5e4d49cd8119ddfae153a41f4cd5deb9d95508cf7ea455bbcad96137eb422d2","family_id":"qf-mod39-subtop-induced-topology","seed":202611083007,"semantic_template":"qf-mod39-subtop-induced-topology/{\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2],[0,2],[0]],\"subset\":[2,3,4]}"},"fields":{"title":"Normalize All Ambient-Open Intersections to Obtain the Relative Topology","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 2]], and subset Y=[2, 3, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Give (subset, induced_topology, open_count, topology_certificate, certificate).","answer":{"kind":"expression","value":"([2, 3, 4], [[], [2], [2, 3], [2, 3, 4]], 4, Yes, all_parent_open_intersections_normalized)","explanation":"Intersect every parent open with Y, identify duplicate intersections and retain the source canonical family order. The empty subspace has the one-open-set topology {empty set}. This finite complete construction does not sample an unspecified ambient topology. The exact result, retaining the requested order, is ([2, 3, 4], [[], [2], [2, 3], [2, 3, 4]], 4, Yes, all_parent_open_intersections_normalized)."},"skills":["Normalize All Ambient-Open Intersections to Obtain the Relative Topology"],"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-20261108-r03-subspace-topology-0008","batch_id":"batch-20261014-advance-20261108-r03-subspace-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"21730fa30d1406e1b75cff4792c975632463f4439218f62bedd2480d3e5454b5","family_id":"qf-mod39-subtop-induced-topology","seed":202611083008,"semantic_template":"qf-mod39-subtop-induced-topology/{\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,2],[0,2],[0]],\"subset\":[2,3,4]}"},"fields":{"title":"Normalize All Ambient-Open Intersections to Obtain the Relative Topology","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 2]], and subset Y=[2, 3, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Give (subset, induced_topology, open_count, topology_certificate, certificate).","answer":{"kind":"expression","value":"([2, 3, 4], [[], [2], [2, 3, 4], [2, 4]], 4, Yes, all_parent_open_intersections_normalized)","explanation":"Intersect every parent open with Y, identify duplicate intersections and retain the source canonical family order. The empty subspace has the one-open-set topology {empty set}. This finite complete construction does not sample an unspecified ambient topology. The exact result, retaining the requested order, is ([2, 3, 4], [[], [2], [2, 3, 4], [2, 4]], 4, Yes, all_parent_open_intersections_normalized)."},"skills":["Normalize All Ambient-Open Intersections to Obtain the Relative Topology"],"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-20261108-r03-subspace-topology-0009","batch_id":"batch-20261014-advance-20261108-r03-subspace-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b36a21761a8eb6633f3efc98e298c22dd1441c28fe64234a1cce144b14d00ebf","family_id":"qf-mod39-subtop-induced-topology","seed":202611083009,"semantic_template":"qf-mod39-subtop-induced-topology/{\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,2,3],[0,2],[0]],\"subset\":[1,2,4]}"},"fields":{"title":"Normalize All Ambient-Open Intersections to Obtain the Relative Topology","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 2], [0, 2, 3]], and subset Y=[1, 2, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Give (subset, induced_topology, open_count, topology_certificate, certificate).","answer":{"kind":"expression","value":"([1, 2, 4], [[], [1, 2], [1, 2, 4], [2]], 4, Yes, all_parent_open_intersections_normalized)","explanation":"Intersect every parent open with Y, identify duplicate intersections and retain the source canonical family order. The empty subspace has the one-open-set topology {empty set}. This finite complete construction does not sample an unspecified ambient topology. The exact result, retaining the requested order, is ([1, 2, 4], [[], [1, 2], [1, 2, 4], [2]], 4, Yes, all_parent_open_intersections_normalized)."},"skills":["Normalize All Ambient-Open Intersections to Obtain the Relative Topology"],"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-20261108-r03-subspace-topology-0010","batch_id":"batch-20261014-advance-20261108-r03-subspace-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5729a65d99ec35e2a2902a5f20bb584b4859dd7a78dcebd6dc477c3c621605ad","family_id":"qf-mod39-subtop-induced-topology","seed":202611083010,"semantic_template":"qf-mod39-subtop-induced-topology/{\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,3],[0,2],[0]],\"subset\":[1,2,4]}"},"fields":{"title":"Normalize All Ambient-Open Intersections to Obtain the Relative Topology","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4]], and subset Y=[1, 2, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Give (subset, induced_topology, open_count, topology_certificate, certificate).","answer":{"kind":"expression","value":"([1, 2, 4], [[], [1, 2, 4], [2], [2, 4]], 4, Yes, all_parent_open_intersections_normalized)","explanation":"Intersect every parent open with Y, identify duplicate intersections and retain the source canonical family order. The empty subspace has the one-open-set topology {empty set}. This finite complete construction does not sample an unspecified ambient topology. The exact result, retaining the requested order, is ([1, 2, 4], [[], [1, 2, 4], [2], [2, 4]], 4, Yes, all_parent_open_intersections_normalized)."},"skills":["Normalize All Ambient-Open Intersections to Obtain the Relative Topology"],"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-20261108-r03-subspace-topology-0011","batch_id":"batch-20261014-advance-20261108-r03-subspace-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4d967aee13634db5185236bde5b2d754bb5c32f425a708ed2f4ff9c5515be32f","family_id":"qf-mod39-subtop-induced-topology","seed":202611083011,"semantic_template":"qf-mod39-subtop-induced-topology/{\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,2,4],[0,2],[0]],\"subset\":[1,2,3]}"},"fields":{"title":"Normalize All Ambient-Open Intersections to Obtain the Relative Topology","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 2], [0, 2, 4]], and subset Y=[1, 2, 3]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Give (subset, induced_topology, open_count, topology_certificate, certificate).","answer":{"kind":"expression","value":"([1, 2, 3], [[], [1, 2], [1, 2, 3], [2]], 4, Yes, all_parent_open_intersections_normalized)","explanation":"Intersect every parent open with Y, identify duplicate intersections and retain the source canonical family order. The empty subspace has the one-open-set topology {empty set}. This finite complete construction does not sample an unspecified ambient topology. The exact result, retaining the requested order, is ([1, 2, 3], [[], [1, 2], [1, 2, 3], [2]], 4, Yes, all_parent_open_intersections_normalized)."},"skills":["Normalize All Ambient-Open Intersections to Obtain the Relative Topology"],"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-20261108-r03-subspace-topology-0012","batch_id":"batch-20261014-advance-20261108-r03-subspace-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"388c96ac2faf055c5c0176dbd54395e6b0b35f9ef2fe13277c59e7ecda29b6bd","family_id":"qf-mod39-subtop-induced-topology","seed":202611083012,"semantic_template":"qf-mod39-subtop-induced-topology/{\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,4],[0,2],[0]],\"subset\":[1,2,3]}"},"fields":{"title":"Normalize All Ambient-Open Intersections to Obtain the Relative Topology","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 2], [0, 2, 3, 4], [0, 2, 4]], and subset Y=[1, 2, 3]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Give (subset, induced_topology, open_count, topology_certificate, certificate).","answer":{"kind":"expression","value":"([1, 2, 3], [[], [1, 2, 3], [2], [2, 3]], 4, Yes, all_parent_open_intersections_normalized)","explanation":"Intersect every parent open with Y, identify duplicate intersections and retain the source canonical family order. The empty subspace has the one-open-set topology {empty set}. This finite complete construction does not sample an unspecified ambient topology. The exact result, retaining the requested order, is ([1, 2, 3], [[], [1, 2, 3], [2], [2, 3]], 4, Yes, all_parent_open_intersections_normalized)."},"skills":["Normalize All Ambient-Open Intersections to Obtain the Relative Topology"],"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-20261108-r03-subspace-topology-0013","batch_id":"batch-20261014-advance-20261108-r03-subspace-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d98c3ecdd510482feceebb305988f034ff0484d282218fed603e0efe81a5bdb3","family_id":"qf-mod39-subtop-induced-topology","seed":202611083013,"semantic_template":"qf-mod39-subtop-induced-topology/{\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,3],[0,3],[0]],\"subset\":[2,3,4]}"},"fields":{"title":"Normalize All Ambient-Open Intersections to Obtain the Relative Topology","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 3], [0, 3]], and subset Y=[2, 3, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Give (subset, induced_topology, open_count, topology_certificate, certificate).","answer":{"kind":"expression","value":"([2, 3, 4], [[], [2, 3], [2, 3, 4], [3]], 4, Yes, all_parent_open_intersections_normalized)","explanation":"Intersect every parent open with Y, identify duplicate intersections and retain the source canonical family order. The empty subspace has the one-open-set topology {empty set}. This finite complete construction does not sample an unspecified ambient topology. The exact result, retaining the requested order, is ([2, 3, 4], [[], [2, 3], [2, 3, 4], [3]], 4, Yes, all_parent_open_intersections_normalized)."},"skills":["Normalize All Ambient-Open Intersections to Obtain the Relative Topology"],"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-20261108-r03-subspace-topology-0014","batch_id":"batch-20261014-advance-20261108-r03-subspace-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"981e0a7d36afac6d4d0575732e452dc7c87b9e211df1b453f37fdb80f7def7dc","family_id":"qf-mod39-subtop-induced-topology","seed":202611083014,"semantic_template":"qf-mod39-subtop-induced-topology/{\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,3],[0,3],[0]],\"subset\":[2,3,4]}"},"fields":{"title":"Normalize All Ambient-Open Intersections to Obtain the Relative Topology","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 3], [0, 1, 3, 4], [0, 3]], and subset Y=[2, 3, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Give (subset, induced_topology, open_count, topology_certificate, certificate).","answer":{"kind":"expression","value":"([2, 3, 4], [[], [2, 3, 4], [3], [3, 4]], 4, Yes, all_parent_open_intersections_normalized)","explanation":"Intersect every parent open with Y, identify duplicate intersections and retain the source canonical family order. The empty subspace has the one-open-set topology {empty set}. This finite complete construction does not sample an unspecified ambient topology. The exact result, retaining the requested order, is ([2, 3, 4], [[], [2, 3, 4], [3], [3, 4]], 4, Yes, all_parent_open_intersections_normalized)."},"skills":["Normalize All Ambient-Open Intersections to Obtain the Relative Topology"],"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-20261108-r03-subspace-topology-0015","batch_id":"batch-20261014-advance-20261108-r03-subspace-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a2b3f1114cc91d9e3ac4f9101a55d0a0b6e8a11baf97940f4fdd6953e244e930","family_id":"qf-mod39-subtop-induced-topology","seed":202611083015,"semantic_template":"qf-mod39-subtop-induced-topology/{\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,2,3],[0,3],[0]],\"subset\":[1,3,4]}"},"fields":{"title":"Normalize All Ambient-Open Intersections to Obtain the Relative Topology","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 2, 3], [0, 3]], and subset Y=[1, 3, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Give (subset, induced_topology, open_count, topology_certificate, certificate).","answer":{"kind":"expression","value":"([1, 3, 4], [[], [1, 3], [1, 3, 4], [3]], 4, Yes, all_parent_open_intersections_normalized)","explanation":"Intersect every parent open with Y, identify duplicate intersections and retain the source canonical family order. The empty subspace has the one-open-set topology {empty set}. This finite complete construction does not sample an unspecified ambient topology. The exact result, retaining the requested order, is ([1, 3, 4], [[], [1, 3], [1, 3, 4], [3]], 4, Yes, all_parent_open_intersections_normalized)."},"skills":["Normalize All Ambient-Open Intersections to Obtain the Relative Topology"],"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-20261108-r03-subspace-topology-0016","batch_id":"batch-20261014-advance-20261108-r03-subspace-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ab65bfdff342d8749cb02b72304d3bdac95c1ffbd48aca3679ec851921b9fb01","family_id":"qf-mod39-subtop-induced-topology","seed":202611083016,"semantic_template":"qf-mod39-subtop-induced-topology/{\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,3],[0,3],[0]],\"subset\":[1,3,4]}"},"fields":{"title":"Normalize All Ambient-Open Intersections to Obtain the Relative Topology","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 2, 3], [0, 2, 3, 4], [0, 3]], and subset Y=[1, 3, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Give (subset, induced_topology, open_count, topology_certificate, certificate).","answer":{"kind":"expression","value":"([1, 3, 4], [[], [1, 3, 4], [3], [3, 4]], 4, Yes, all_parent_open_intersections_normalized)","explanation":"Intersect every parent open with Y, identify duplicate intersections and retain the source canonical family order. The empty subspace has the one-open-set topology {empty set}. This finite complete construction does not sample an unspecified ambient topology. The exact result, retaining the requested order, is ([1, 3, 4], [[], [1, 3, 4], [3], [3, 4]], 4, Yes, all_parent_open_intersections_normalized)."},"skills":["Normalize All Ambient-Open Intersections to Obtain the Relative Topology"],"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-20261108-r03-subspace-topology-0017","batch_id":"batch-20261014-advance-20261108-r03-subspace-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9f2711283e1897c1949aee8568fd9ed790c4076aa848b032a39baff6ce29f872","family_id":"qf-mod39-subtop-induced-topology","seed":202611083017,"semantic_template":"qf-mod39-subtop-induced-topology/{\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,3,4],[0,3],[0]],\"subset\":[1,2,3]}"},"fields":{"title":"Normalize All Ambient-Open Intersections to Obtain the Relative Topology","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 3, 4], [0, 3], [0, 3, 4]], and subset Y=[1, 2, 3]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Give (subset, induced_topology, open_count, topology_certificate, certificate).","answer":{"kind":"expression","value":"([1, 2, 3], [[], [1, 2, 3], [1, 3], [3]], 4, Yes, all_parent_open_intersections_normalized)","explanation":"Intersect every parent open with Y, identify duplicate intersections and retain the source canonical family order. The empty subspace has the one-open-set topology {empty set}. This finite complete construction does not sample an unspecified ambient topology. The exact result, retaining the requested order, is ([1, 2, 3], [[], [1, 2, 3], [1, 3], [3]], 4, Yes, all_parent_open_intersections_normalized)."},"skills":["Normalize All Ambient-Open Intersections to Obtain the Relative Topology"],"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-20261108-r03-subspace-topology-0018","batch_id":"batch-20261014-advance-20261108-r03-subspace-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"46b520bda0e6eea397b9b0e49ac0d2b2b895563f97879bc38f9eee8303b26d36","family_id":"qf-mod39-subtop-induced-topology","seed":202611083018,"semantic_template":"qf-mod39-subtop-induced-topology/{\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,3,4],[0,3],[0]],\"subset\":[1,2,3]}"},"fields":{"title":"Normalize All Ambient-Open Intersections to Obtain the Relative Topology","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 2, 3, 4], [0, 3], [0, 3, 4]], and subset Y=[1, 2, 3]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Give (subset, induced_topology, open_count, topology_certificate, certificate).","answer":{"kind":"expression","value":"([1, 2, 3], [[], [1, 2, 3], [2, 3], [3]], 4, Yes, all_parent_open_intersections_normalized)","explanation":"Intersect every parent open with Y, identify duplicate intersections and retain the source canonical family order. The empty subspace has the one-open-set topology {empty set}. This finite complete construction does not sample an unspecified ambient topology. The exact result, retaining the requested order, is ([1, 2, 3], [[], [1, 2, 3], [2, 3], [3]], 4, Yes, all_parent_open_intersections_normalized)."},"skills":["Normalize All Ambient-Open Intersections to Obtain the Relative Topology"],"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-20261108-r03-subspace-topology-0019","batch_id":"batch-20261014-advance-20261108-r03-subspace-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"207bbba2c37fecd68bcd929a8038280e594b983d39f4b0d13d40f4ff59470bef","family_id":"qf-mod39-subtop-induced-topology","seed":202611083019,"semantic_template":"qf-mod39-subtop-induced-topology/{\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,4],[0,4],[0]],\"subset\":[2,3,4]}"},"fields":{"title":"Normalize All Ambient-Open Intersections to Obtain the Relative Topology","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 4], [0, 4]], and subset Y=[2, 3, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Give (subset, induced_topology, open_count, topology_certificate, certificate).","answer":{"kind":"expression","value":"([2, 3, 4], [[], [2, 3, 4], [2, 4], [4]], 4, Yes, all_parent_open_intersections_normalized)","explanation":"Intersect every parent open with Y, identify duplicate intersections and retain the source canonical family order. The empty subspace has the one-open-set topology {empty set}. This finite complete construction does not sample an unspecified ambient topology. The exact result, retaining the requested order, is ([2, 3, 4], [[], [2, 3, 4], [2, 4], [4]], 4, Yes, all_parent_open_intersections_normalized)."},"skills":["Normalize All Ambient-Open Intersections to Obtain the Relative Topology"],"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-20261108-r03-subspace-topology-0020","batch_id":"batch-20261014-advance-20261108-r03-subspace-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6ce4ada015192c72d6bb3fefb4c29184760b65a648d6a4f1768eb220ef62ccd4","family_id":"qf-mod39-subtop-induced-topology","seed":202611083020,"semantic_template":"qf-mod39-subtop-induced-topology/{\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,4],[0,4],[0]],\"subset\":[2,3,4]}"},"fields":{"title":"Normalize All Ambient-Open Intersections to Obtain the Relative Topology","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 3, 4], [0, 1, 4], [0, 4]], and subset Y=[2, 3, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Give (subset, induced_topology, open_count, topology_certificate, certificate).","answer":{"kind":"expression","value":"([2, 3, 4], [[], [2, 3, 4], [3, 4], [4]], 4, Yes, all_parent_open_intersections_normalized)","explanation":"Intersect every parent open with Y, identify duplicate intersections and retain the source canonical family order. The empty subspace has the one-open-set topology {empty set}. This finite complete construction does not sample an unspecified ambient topology. The exact result, retaining the requested order, is ([2, 3, 4], [[], [2, 3, 4], [3, 4], [4]], 4, Yes, all_parent_open_intersections_normalized)."},"skills":["Normalize All Ambient-Open Intersections to Obtain the Relative Topology"],"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-20261108-r03-subspace-topology-0021","batch_id":"batch-20261014-advance-20261108-r03-subspace-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7593654289a456ce3be67f1b9bff01fefdeec68c8cf48715ca8c4b535f890683","family_id":"qf-mod39-subtop-induced-topology","seed":202611083021,"semantic_template":"qf-mod39-subtop-induced-topology/{\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,2,4],[0,4],[0]],\"subset\":[1,3,4]}"},"fields":{"title":"Normalize All Ambient-Open Intersections to Obtain the Relative Topology","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 2, 4], [0, 4]], and subset Y=[1, 3, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Give (subset, induced_topology, open_count, topology_certificate, certificate).","answer":{"kind":"expression","value":"([1, 3, 4], [[], [1, 3, 4], [1, 4], [4]], 4, Yes, all_parent_open_intersections_normalized)","explanation":"Intersect every parent open with Y, identify duplicate intersections and retain the source canonical family order. The empty subspace has the one-open-set topology {empty set}. This finite complete construction does not sample an unspecified ambient topology. The exact result, retaining the requested order, is ([1, 3, 4], [[], [1, 3, 4], [1, 4], [4]], 4, Yes, all_parent_open_intersections_normalized)."},"skills":["Normalize All Ambient-Open Intersections to Obtain the Relative Topology"],"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-20261108-r03-subspace-topology-0022","batch_id":"batch-20261014-advance-20261108-r03-subspace-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9c92e9880db2f253ad968c67d7610e675047bf3555c43ed09b506b1d2b4cb689","family_id":"qf-mod39-subtop-induced-topology","seed":202611083022,"semantic_template":"qf-mod39-subtop-induced-topology/{\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,4],[0,4],[0]],\"subset\":[1,3,4]}"},"fields":{"title":"Normalize All Ambient-Open Intersections to Obtain the Relative Topology","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 2, 3, 4], [0, 2, 4], [0, 4]], and subset Y=[1, 3, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Give (subset, induced_topology, open_count, topology_certificate, certificate).","answer":{"kind":"expression","value":"([1, 3, 4], [[], [1, 3, 4], [3, 4], [4]], 4, Yes, all_parent_open_intersections_normalized)","explanation":"Intersect every parent open with Y, identify duplicate intersections and retain the source canonical family order. The empty subspace has the one-open-set topology {empty set}. This finite complete construction does not sample an unspecified ambient topology. The exact result, retaining the requested order, is ([1, 3, 4], [[], [1, 3, 4], [3, 4], [4]], 4, Yes, all_parent_open_intersections_normalized)."},"skills":["Normalize All Ambient-Open Intersections to Obtain the Relative Topology"],"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-20261108-r03-subspace-topology-0023","batch_id":"batch-20261014-advance-20261108-r03-subspace-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"73ebff1355b6edda9ea073db3edc2b9eaa15eb350c83f1fd0cc55b79bece50f1","family_id":"qf-mod39-subtop-induced-topology","seed":202611083023,"semantic_template":"qf-mod39-subtop-induced-topology/{\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,3,4],[0,4],[0]],\"subset\":[1,2,4]}"},"fields":{"title":"Normalize All Ambient-Open Intersections to Obtain the Relative Topology","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 3, 4], [0, 3, 4], [0, 4]], and subset Y=[1, 2, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Give (subset, induced_topology, open_count, topology_certificate, certificate).","answer":{"kind":"expression","value":"([1, 2, 4], [[], [1, 2, 4], [1, 4], [4]], 4, Yes, all_parent_open_intersections_normalized)","explanation":"Intersect every parent open with Y, identify duplicate intersections and retain the source canonical family order. The empty subspace has the one-open-set topology {empty set}. This finite complete construction does not sample an unspecified ambient topology. The exact result, retaining the requested order, is ([1, 2, 4], [[], [1, 2, 4], [1, 4], [4]], 4, Yes, all_parent_open_intersections_normalized)."},"skills":["Normalize All Ambient-Open Intersections to Obtain the Relative Topology"],"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-20261108-r03-subspace-topology-0024","batch_id":"batch-20261014-advance-20261108-r03-subspace-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9f35034989b057513b3ed5be799ecc58bb9f3c6b4a4f230b3ece95bc074f90dc","family_id":"qf-mod39-subtop-induced-topology","seed":202611083024,"semantic_template":"qf-mod39-subtop-induced-topology/{\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,3,4],[0,4],[0]],\"subset\":[1,2,4]}"},"fields":{"title":"Normalize All Ambient-Open Intersections to Obtain the Relative Topology","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 2, 3, 4], [0, 3, 4], [0, 4]], and subset Y=[1, 2, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Give (subset, induced_topology, open_count, topology_certificate, certificate).","answer":{"kind":"expression","value":"([1, 2, 4], [[], [1, 2, 4], [2, 4], [4]], 4, Yes, all_parent_open_intersections_normalized)","explanation":"Intersect every parent open with Y, identify duplicate intersections and retain the source canonical family order. The empty subspace has the one-open-set topology {empty set}. This finite complete construction does not sample an unspecified ambient topology. The exact result, retaining the requested order, is ([1, 2, 4], [[], [1, 2, 4], [2, 4], [4]], 4, Yes, all_parent_open_intersections_normalized)."},"skills":["Normalize All Ambient-Open Intersections to Obtain the Relative Topology"],"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-20261108-r03-subspace-topology-0025","batch_id":"batch-20261014-advance-20261108-r03-subspace-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"81f8b1a7d03c566fc4fc2930b19852e61a6c0aa91acbb0f53699465bec6fa783","family_id":"qf-mod39-subtop-induced-topology","seed":202611083025,"semantic_template":"qf-mod39-subtop-induced-topology/{\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2],[0,1],[1]],\"subset\":[0,3,4]}"},"fields":{"title":"Normalize All Ambient-Open Intersections to Obtain the Relative Topology","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [1]], and subset Y=[0, 3, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Give (subset, induced_topology, open_count, topology_certificate, certificate).","answer":{"kind":"expression","value":"([0, 3, 4], [[], [0], [0, 3], [0, 3, 4]], 4, Yes, all_parent_open_intersections_normalized)","explanation":"Intersect every parent open with Y, identify duplicate intersections and retain the source canonical family order. The empty subspace has the one-open-set topology {empty set}. This finite complete construction does not sample an unspecified ambient topology. The exact result, retaining the requested order, is ([0, 3, 4], [[], [0], [0, 3], [0, 3, 4]], 4, Yes, all_parent_open_intersections_normalized)."},"skills":["Normalize All Ambient-Open Intersections to Obtain the Relative Topology"],"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-20261108-r03-subspace-topology-0026","batch_id":"batch-20261014-advance-20261108-r03-subspace-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d5ea15174bbbd4b9f221a3371292aeaf1d7ca8f0b82058bf5d203991a2f6c673","family_id":"qf-mod39-subtop-relative-open-audit","seed":202611083026,"semantic_template":"qf-mod39-subtop-relative-open-audit/{\"candidate\":[1],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2],[0,1],[0]],\"subset\":[1,3,4]}"},"fields":{"title":"A Relative Open May Need an Ambient Open Extending beyond Y","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]], and subset Y=[1, 3, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[1] is a duplicate-free subset of Y. Give (candidate, relative_open, parent_open_witnesses, boundary).","answer":{"kind":"expression","value":"([1], Yes, [[0, 1], [0, 1, 2]], relative_open_need_not_be_ambient_open)","explanation":"Check membership in the induced topology, then list every ambient open U with U∩Y equal to the candidate. U need not be contained in Y, and relative openness need not be ambient openness. The exact result, retaining the requested order, is ([1], Yes, [[0, 1], [0, 1, 2]], relative_open_need_not_be_ambient_open)."},"skills":["A Relative Open May Need an Ambient Open Extending beyond Y"],"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-20261108-r03-subspace-topology-0027","batch_id":"batch-20261014-advance-20261108-r03-subspace-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"332f826d46b245045a5dccf6ac0e4522d2ebdd7145cedc9afd22179537e51041","family_id":"qf-mod39-subtop-relative-open-audit","seed":202611083027,"semantic_template":"qf-mod39-subtop-relative-open-audit/{\"candidate\":[1],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,2],[0,1],[0]],\"subset\":[1,3,4]}"},"fields":{"title":"A Relative Open May Need an Ambient Open Extending beyond Y","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3, 4], [0, 1, 2, 4]], and subset Y=[1, 3, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[1] is a duplicate-free subset of Y. Give (candidate, relative_open, parent_open_witnesses, boundary).","answer":{"kind":"expression","value":"([1], Yes, [[0, 1], [0, 1, 2]], relative_open_need_not_be_ambient_open)","explanation":"Check membership in the induced topology, then list every ambient open U with U∩Y equal to the candidate. U need not be contained in Y, and relative openness need not be ambient openness. The exact result, retaining the requested order, is ([1], Yes, [[0, 1], [0, 1, 2]], relative_open_need_not_be_ambient_open)."},"skills":["A Relative Open May Need an Ambient Open Extending beyond Y"],"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-20261108-r03-subspace-topology-0028","batch_id":"batch-20261014-advance-20261108-r03-subspace-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b06ac2cd7e726858b5dfc84ad7af14882faa404381de7888f2924bab75d193a1","family_id":"qf-mod39-subtop-relative-open-audit","seed":202611083028,"semantic_template":"qf-mod39-subtop-relative-open-audit/{\"candidate\":[1],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,3],[0,1],[0]],\"subset\":[1,2,4]}"},"fields":{"title":"A Relative Open May Need an Ambient Open Extending beyond Y","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 3]], and subset Y=[1, 2, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[1] is a duplicate-free subset of Y. Give (candidate, relative_open, parent_open_witnesses, boundary).","answer":{"kind":"expression","value":"([1], Yes, [[0, 1], [0, 1, 3]], relative_open_need_not_be_ambient_open)","explanation":"Check membership in the induced topology, then list every ambient open U with U∩Y equal to the candidate. U need not be contained in Y, and relative openness need not be ambient openness. The exact result, retaining the requested order, is ([1], Yes, [[0, 1], [0, 1, 3]], relative_open_need_not_be_ambient_open)."},"skills":["A Relative Open May Need an Ambient Open Extending beyond Y"],"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-20261108-r03-subspace-topology-0029","batch_id":"batch-20261014-advance-20261108-r03-subspace-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e0060ebf32fc8fcd6743ee12c8af3195a61861212861828055d848bff2b0ea67","family_id":"qf-mod39-subtop-relative-open-audit","seed":202611083029,"semantic_template":"qf-mod39-subtop-relative-open-audit/{\"candidate\":[1],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,3],[0,1],[0]],\"subset\":[1,2,4]}"},"fields":{"title":"A Relative Open May Need an Ambient Open Extending beyond Y","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1], [0, 1, 2, 3, 4], [0, 1, 3], [0, 1, 3, 4]], and subset Y=[1, 2, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[1] is a duplicate-free subset of Y. Give (candidate, relative_open, parent_open_witnesses, boundary).","answer":{"kind":"expression","value":"([1], Yes, [[0, 1], [0, 1, 3]], relative_open_need_not_be_ambient_open)","explanation":"Check membership in the induced topology, then list every ambient open U with U∩Y equal to the candidate. U need not be contained in Y, and relative openness need not be ambient openness. The exact result, retaining the requested order, is ([1], Yes, [[0, 1], [0, 1, 3]], relative_open_need_not_be_ambient_open)."},"skills":["A Relative Open May Need an Ambient Open Extending beyond Y"],"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-20261108-r03-subspace-topology-0030","batch_id":"batch-20261014-advance-20261108-r03-subspace-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"71a863daa27e674f5d52c4e50f112eb44399270f53f2e53d27652d98390e7c4e","family_id":"qf-mod39-subtop-relative-open-audit","seed":202611083030,"semantic_template":"qf-mod39-subtop-relative-open-audit/{\"candidate\":[1],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,4],[0,1],[0]],\"subset\":[1,2,3]}"},"fields":{"title":"A Relative Open May Need an Ambient Open Extending beyond Y","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 4]], and subset Y=[1, 2, 3]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[1] is a duplicate-free subset of Y. Give (candidate, relative_open, parent_open_witnesses, boundary).","answer":{"kind":"expression","value":"([1], Yes, [[0, 1], [0, 1, 4]], relative_open_need_not_be_ambient_open)","explanation":"Check membership in the induced topology, then list every ambient open U with U∩Y equal to the candidate. U need not be contained in Y, and relative openness need not be ambient openness. The exact result, retaining the requested order, is ([1], Yes, [[0, 1], [0, 1, 4]], relative_open_need_not_be_ambient_open)."},"skills":["A Relative Open May Need an Ambient Open Extending beyond Y"],"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-20261108-r03-subspace-topology-0031","batch_id":"batch-20261014-advance-20261108-r03-subspace-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7e7a453ca015699bb4a88c1a9bbcde303f089659377e821a6c3f34c06e246d93","family_id":"qf-mod39-subtop-relative-open-audit","seed":202611083031,"semantic_template":"qf-mod39-subtop-relative-open-audit/{\"candidate\":[1],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,4],[0,1],[0]],\"subset\":[1,2,3]}"},"fields":{"title":"A Relative Open May Need an Ambient Open Extending beyond Y","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1], [0, 1, 2, 3, 4], [0, 1, 3, 4], [0, 1, 4]], and subset Y=[1, 2, 3]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[1] is a duplicate-free subset of Y. Give (candidate, relative_open, parent_open_witnesses, boundary).","answer":{"kind":"expression","value":"([1], Yes, [[0, 1], [0, 1, 4]], relative_open_need_not_be_ambient_open)","explanation":"Check membership in the induced topology, then list every ambient open U with U∩Y equal to the candidate. U need not be contained in Y, and relative openness need not be ambient openness. The exact result, retaining the requested order, is ([1], Yes, [[0, 1], [0, 1, 4]], relative_open_need_not_be_ambient_open)."},"skills":["A Relative Open May Need an Ambient Open Extending beyond Y"],"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-20261108-r03-subspace-topology-0032","batch_id":"batch-20261014-advance-20261108-r03-subspace-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3d6519f4879d3bf05d4868c74088945b5f95c8db4a01940a921616e6d81d8991","family_id":"qf-mod39-subtop-relative-open-audit","seed":202611083032,"semantic_template":"qf-mod39-subtop-relative-open-audit/{\"candidate\":[2],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2],[0,2],[0]],\"subset\":[2,3,4]}"},"fields":{"title":"A Relative Open May Need an Ambient Open Extending beyond Y","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 2]], and subset Y=[2, 3, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[2] is a duplicate-free subset of Y. Give (candidate, relative_open, parent_open_witnesses, boundary).","answer":{"kind":"expression","value":"([2], Yes, [[0, 1, 2], [0, 2]], relative_open_need_not_be_ambient_open)","explanation":"Check membership in the induced topology, then list every ambient open U with U∩Y equal to the candidate. U need not be contained in Y, and relative openness need not be ambient openness. The exact result, retaining the requested order, is ([2], Yes, [[0, 1, 2], [0, 2]], relative_open_need_not_be_ambient_open)."},"skills":["A Relative Open May Need an Ambient Open Extending beyond Y"],"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-20261108-r03-subspace-topology-0033","batch_id":"batch-20261014-advance-20261108-r03-subspace-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d47677bd6ed663a0a1e1a1145f6ae493bdb6f73d10e6eb9b676888c43219db9b","family_id":"qf-mod39-subtop-relative-open-audit","seed":202611083033,"semantic_template":"qf-mod39-subtop-relative-open-audit/{\"candidate\":[2],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,2],[0,2],[0]],\"subset\":[2,3,4]}"},"fields":{"title":"A Relative Open May Need an Ambient Open Extending beyond Y","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 2]], and subset Y=[2, 3, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[2] is a duplicate-free subset of Y. Give (candidate, relative_open, parent_open_witnesses, boundary).","answer":{"kind":"expression","value":"([2], Yes, [[0, 1, 2], [0, 2]], relative_open_need_not_be_ambient_open)","explanation":"Check membership in the induced topology, then list every ambient open U with U∩Y equal to the candidate. U need not be contained in Y, and relative openness need not be ambient openness. The exact result, retaining the requested order, is ([2], Yes, [[0, 1, 2], [0, 2]], relative_open_need_not_be_ambient_open)."},"skills":["A Relative Open May Need an Ambient Open Extending beyond Y"],"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-20261108-r03-subspace-topology-0034","batch_id":"batch-20261014-advance-20261108-r03-subspace-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1c6d36fa7fe10e02f16b75b2422b840b1bceced512959a8a4095be47e8395afb","family_id":"qf-mod39-subtop-relative-open-audit","seed":202611083034,"semantic_template":"qf-mod39-subtop-relative-open-audit/{\"candidate\":[2],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,2,3],[0,2],[0]],\"subset\":[1,2,4]}"},"fields":{"title":"A Relative Open May Need an Ambient Open Extending beyond Y","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 2], [0, 2, 3]], and subset Y=[1, 2, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[2] is a duplicate-free subset of Y. Give (candidate, relative_open, parent_open_witnesses, boundary).","answer":{"kind":"expression","value":"([2], Yes, [[0, 2], [0, 2, 3]], relative_open_need_not_be_ambient_open)","explanation":"Check membership in the induced topology, then list every ambient open U with U∩Y equal to the candidate. U need not be contained in Y, and relative openness need not be ambient openness. The exact result, retaining the requested order, is ([2], Yes, [[0, 2], [0, 2, 3]], relative_open_need_not_be_ambient_open)."},"skills":["A Relative Open May Need an Ambient Open Extending beyond Y"],"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-20261108-r03-subspace-topology-0035","batch_id":"batch-20261014-advance-20261108-r03-subspace-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f819d16f74183f4912ee35fa9eff180b86c746ada1e8eb47570f2b28404edf29","family_id":"qf-mod39-subtop-relative-open-audit","seed":202611083035,"semantic_template":"qf-mod39-subtop-relative-open-audit/{\"candidate\":[2],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,3],[0,2],[0]],\"subset\":[1,2,4]}"},"fields":{"title":"A Relative Open May Need an Ambient Open Extending beyond Y","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4]], and subset Y=[1, 2, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[2] is a duplicate-free subset of Y. Give (candidate, relative_open, parent_open_witnesses, boundary).","answer":{"kind":"expression","value":"([2], Yes, [[0, 2], [0, 2, 3]], relative_open_need_not_be_ambient_open)","explanation":"Check membership in the induced topology, then list every ambient open U with U∩Y equal to the candidate. U need not be contained in Y, and relative openness need not be ambient openness. The exact result, retaining the requested order, is ([2], Yes, [[0, 2], [0, 2, 3]], relative_open_need_not_be_ambient_open)."},"skills":["A Relative Open May Need an Ambient Open Extending beyond Y"],"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-20261108-r03-subspace-topology-0036","batch_id":"batch-20261014-advance-20261108-r03-subspace-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"21e2be11a0533e64879cd6ce390f1eff92ac0fa42d8ab7e7ab273e09718a077d","family_id":"qf-mod39-subtop-relative-open-audit","seed":202611083036,"semantic_template":"qf-mod39-subtop-relative-open-audit/{\"candidate\":[2],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,2,4],[0,2],[0]],\"subset\":[1,2,3]}"},"fields":{"title":"A Relative Open May Need an Ambient Open Extending beyond Y","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 2], [0, 2, 4]], and subset Y=[1, 2, 3]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[2] is a duplicate-free subset of Y. Give (candidate, relative_open, parent_open_witnesses, boundary).","answer":{"kind":"expression","value":"([2], Yes, [[0, 2], [0, 2, 4]], relative_open_need_not_be_ambient_open)","explanation":"Check membership in the induced topology, then list every ambient open U with U∩Y equal to the candidate. U need not be contained in Y, and relative openness need not be ambient openness. The exact result, retaining the requested order, is ([2], Yes, [[0, 2], [0, 2, 4]], relative_open_need_not_be_ambient_open)."},"skills":["A Relative Open May Need an Ambient Open Extending beyond Y"],"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-20261108-r03-subspace-topology-0037","batch_id":"batch-20261014-advance-20261108-r03-subspace-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"da3326e301d4bd22ec9fd64e60246d77cf20cff5d9d9c4e513e76e517af62b16","family_id":"qf-mod39-subtop-relative-open-audit","seed":202611083037,"semantic_template":"qf-mod39-subtop-relative-open-audit/{\"candidate\":[2],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,4],[0,2],[0]],\"subset\":[1,2,3]}"},"fields":{"title":"A Relative Open May Need an Ambient Open Extending beyond Y","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 2], [0, 2, 3, 4], [0, 2, 4]], and subset Y=[1, 2, 3]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[2] is a duplicate-free subset of Y. Give (candidate, relative_open, parent_open_witnesses, boundary).","answer":{"kind":"expression","value":"([2], Yes, [[0, 2], [0, 2, 4]], relative_open_need_not_be_ambient_open)","explanation":"Check membership in the induced topology, then list every ambient open U with U∩Y equal to the candidate. U need not be contained in Y, and relative openness need not be ambient openness. The exact result, retaining the requested order, is ([2], Yes, [[0, 2], [0, 2, 4]], relative_open_need_not_be_ambient_open)."},"skills":["A Relative Open May Need an Ambient Open Extending beyond Y"],"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-20261108-r03-subspace-topology-0038","batch_id":"batch-20261014-advance-20261108-r03-subspace-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1491c5e05365e04ed20fad712caa598bab5956a3981658a3d043654ae1e6d54c","family_id":"qf-mod39-subtop-relative-open-audit","seed":202611083038,"semantic_template":"qf-mod39-subtop-relative-open-audit/{\"candidate\":[3],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,3],[0,3],[0]],\"subset\":[2,3,4]}"},"fields":{"title":"A Relative Open May Need an Ambient Open Extending beyond Y","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 3], [0, 3]], and subset Y=[2, 3, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[3] is a duplicate-free subset of Y. Give (candidate, relative_open, parent_open_witnesses, boundary).","answer":{"kind":"expression","value":"([3], Yes, [[0, 1, 3], [0, 3]], relative_open_need_not_be_ambient_open)","explanation":"Check membership in the induced topology, then list every ambient open U with U∩Y equal to the candidate. U need not be contained in Y, and relative openness need not be ambient openness. The exact result, retaining the requested order, is ([3], Yes, [[0, 1, 3], [0, 3]], relative_open_need_not_be_ambient_open)."},"skills":["A Relative Open May Need an Ambient Open Extending beyond Y"],"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-20261108-r03-subspace-topology-0039","batch_id":"batch-20261014-advance-20261108-r03-subspace-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"af03cdb124716aefce47f7c3a74765d1ce2703fe3e338bdb5c20f68366072ca8","family_id":"qf-mod39-subtop-relative-open-audit","seed":202611083039,"semantic_template":"qf-mod39-subtop-relative-open-audit/{\"candidate\":[3],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,3],[0,3],[0]],\"subset\":[2,3,4]}"},"fields":{"title":"A Relative Open May Need an Ambient Open Extending beyond Y","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 3], [0, 1, 3, 4], [0, 3]], and subset Y=[2, 3, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[3] is a duplicate-free subset of Y. Give (candidate, relative_open, parent_open_witnesses, boundary).","answer":{"kind":"expression","value":"([3], Yes, [[0, 1, 3], [0, 3]], relative_open_need_not_be_ambient_open)","explanation":"Check membership in the induced topology, then list every ambient open U with U∩Y equal to the candidate. U need not be contained in Y, and relative openness need not be ambient openness. The exact result, retaining the requested order, is ([3], Yes, [[0, 1, 3], [0, 3]], relative_open_need_not_be_ambient_open)."},"skills":["A Relative Open May Need an Ambient Open Extending beyond Y"],"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-20261108-r03-subspace-topology-0040","batch_id":"batch-20261014-advance-20261108-r03-subspace-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f3d95ffcb6b05a8e2ea04a74dfe026a55c254c702a128e9550c385afdcc1008b","family_id":"qf-mod39-subtop-relative-open-audit","seed":202611083040,"semantic_template":"qf-mod39-subtop-relative-open-audit/{\"candidate\":[3],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,2,3],[0,3],[0]],\"subset\":[1,3,4]}"},"fields":{"title":"A Relative Open May Need an Ambient Open Extending beyond Y","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 2, 3], [0, 3]], and subset Y=[1, 3, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[3] is a duplicate-free subset of Y. Give (candidate, relative_open, parent_open_witnesses, boundary).","answer":{"kind":"expression","value":"([3], Yes, [[0, 2, 3], [0, 3]], relative_open_need_not_be_ambient_open)","explanation":"Check membership in the induced topology, then list every ambient open U with U∩Y equal to the candidate. U need not be contained in Y, and relative openness need not be ambient openness. The exact result, retaining the requested order, is ([3], Yes, [[0, 2, 3], [0, 3]], relative_open_need_not_be_ambient_open)."},"skills":["A Relative Open May Need an Ambient Open Extending beyond Y"],"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-20261108-r03-subspace-topology-0041","batch_id":"batch-20261014-advance-20261108-r03-subspace-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7bae0ad37221cc73749233c64a95bef6ca1ff3115f971992edbe2160dd8ee9f8","family_id":"qf-mod39-subtop-relative-open-audit","seed":202611083041,"semantic_template":"qf-mod39-subtop-relative-open-audit/{\"candidate\":[3],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,3],[0,3],[0]],\"subset\":[1,3,4]}"},"fields":{"title":"A Relative Open May Need an Ambient Open Extending beyond Y","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 2, 3], [0, 2, 3, 4], [0, 3]], and subset Y=[1, 3, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[3] is a duplicate-free subset of Y. Give (candidate, relative_open, parent_open_witnesses, boundary).","answer":{"kind":"expression","value":"([3], Yes, [[0, 2, 3], [0, 3]], relative_open_need_not_be_ambient_open)","explanation":"Check membership in the induced topology, then list every ambient open U with U∩Y equal to the candidate. U need not be contained in Y, and relative openness need not be ambient openness. The exact result, retaining the requested order, is ([3], Yes, [[0, 2, 3], [0, 3]], relative_open_need_not_be_ambient_open)."},"skills":["A Relative Open May Need an Ambient Open Extending beyond Y"],"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-20261108-r03-subspace-topology-0042","batch_id":"batch-20261014-advance-20261108-r03-subspace-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c3ece84b79f56eaf93e6d4e64944c779b5e577ca9cada2d14b6c38a87a2df583","family_id":"qf-mod39-subtop-relative-open-audit","seed":202611083042,"semantic_template":"qf-mod39-subtop-relative-open-audit/{\"candidate\":[3],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,3,4],[0,3],[0]],\"subset\":[1,2,3]}"},"fields":{"title":"A Relative Open May Need an Ambient Open Extending beyond Y","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 3, 4], [0, 3], [0, 3, 4]], and subset Y=[1, 2, 3]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[3] is a duplicate-free subset of Y. Give (candidate, relative_open, parent_open_witnesses, boundary).","answer":{"kind":"expression","value":"([3], Yes, [[0, 3], [0, 3, 4]], relative_open_need_not_be_ambient_open)","explanation":"Check membership in the induced topology, then list every ambient open U with U∩Y equal to the candidate. U need not be contained in Y, and relative openness need not be ambient openness. The exact result, retaining the requested order, is ([3], Yes, [[0, 3], [0, 3, 4]], relative_open_need_not_be_ambient_open)."},"skills":["A Relative Open May Need an Ambient Open Extending beyond Y"],"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-20261108-r03-subspace-topology-0043","batch_id":"batch-20261014-advance-20261108-r03-subspace-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"417ba5ba1875a353d7bc17556519c57b722b56549bfd70b71e4ffe2564575c62","family_id":"qf-mod39-subtop-relative-open-audit","seed":202611083043,"semantic_template":"qf-mod39-subtop-relative-open-audit/{\"candidate\":[3],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,3,4],[0,3],[0]],\"subset\":[1,2,3]}"},"fields":{"title":"A Relative Open May Need an Ambient Open Extending beyond Y","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 2, 3, 4], [0, 3], [0, 3, 4]], and subset Y=[1, 2, 3]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[3] is a duplicate-free subset of Y. Give (candidate, relative_open, parent_open_witnesses, boundary).","answer":{"kind":"expression","value":"([3], Yes, [[0, 3], [0, 3, 4]], relative_open_need_not_be_ambient_open)","explanation":"Check membership in the induced topology, then list every ambient open U with U∩Y equal to the candidate. U need not be contained in Y, and relative openness need not be ambient openness. The exact result, retaining the requested order, is ([3], Yes, [[0, 3], [0, 3, 4]], relative_open_need_not_be_ambient_open)."},"skills":["A Relative Open May Need an Ambient Open Extending beyond Y"],"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-20261108-r03-subspace-topology-0044","batch_id":"batch-20261014-advance-20261108-r03-subspace-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f030c512e7482508f9a76433076713c121138d3906145efe5b5ed6663b866be0","family_id":"qf-mod39-subtop-relative-open-audit","seed":202611083044,"semantic_template":"qf-mod39-subtop-relative-open-audit/{\"candidate\":[4],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,4],[0,4],[0]],\"subset\":[2,3,4]}"},"fields":{"title":"A Relative Open May Need an Ambient Open Extending beyond Y","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 4], [0, 4]], and subset Y=[2, 3, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[4] is a duplicate-free subset of Y. Give (candidate, relative_open, parent_open_witnesses, boundary).","answer":{"kind":"expression","value":"([4], Yes, [[0, 1, 4], [0, 4]], relative_open_need_not_be_ambient_open)","explanation":"Check membership in the induced topology, then list every ambient open U with U∩Y equal to the candidate. U need not be contained in Y, and relative openness need not be ambient openness. The exact result, retaining the requested order, is ([4], Yes, [[0, 1, 4], [0, 4]], relative_open_need_not_be_ambient_open)."},"skills":["A Relative Open May Need an Ambient Open Extending beyond Y"],"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-20261108-r03-subspace-topology-0045","batch_id":"batch-20261014-advance-20261108-r03-subspace-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7e45488f7ba81f2dfbc8ff451a3903fc4d0f0fe1c9891e9b8c5fe414e5ac3568","family_id":"qf-mod39-subtop-relative-open-audit","seed":202611083045,"semantic_template":"qf-mod39-subtop-relative-open-audit/{\"candidate\":[4],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,4],[0,4],[0]],\"subset\":[2,3,4]}"},"fields":{"title":"A Relative Open May Need an Ambient Open Extending beyond Y","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 3, 4], [0, 1, 4], [0, 4]], and subset Y=[2, 3, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[4] is a duplicate-free subset of Y. Give (candidate, relative_open, parent_open_witnesses, boundary).","answer":{"kind":"expression","value":"([4], Yes, [[0, 1, 4], [0, 4]], relative_open_need_not_be_ambient_open)","explanation":"Check membership in the induced topology, then list every ambient open U with U∩Y equal to the candidate. U need not be contained in Y, and relative openness need not be ambient openness. The exact result, retaining the requested order, is ([4], Yes, [[0, 1, 4], [0, 4]], relative_open_need_not_be_ambient_open)."},"skills":["A Relative Open May Need an Ambient Open Extending beyond Y"],"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-20261108-r03-subspace-topology-0046","batch_id":"batch-20261014-advance-20261108-r03-subspace-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6f4a1dedc3e61c0dd11ffc6c226611219057b4e0c02f4399d3ffae2be1b650ee","family_id":"qf-mod39-subtop-relative-open-audit","seed":202611083046,"semantic_template":"qf-mod39-subtop-relative-open-audit/{\"candidate\":[4],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,2,4],[0,4],[0]],\"subset\":[1,3,4]}"},"fields":{"title":"A Relative Open May Need an Ambient Open Extending beyond Y","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 2, 4], [0, 4]], and subset Y=[1, 3, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[4] is a duplicate-free subset of Y. Give (candidate, relative_open, parent_open_witnesses, boundary).","answer":{"kind":"expression","value":"([4], Yes, [[0, 2, 4], [0, 4]], relative_open_need_not_be_ambient_open)","explanation":"Check membership in the induced topology, then list every ambient open U with U∩Y equal to the candidate. U need not be contained in Y, and relative openness need not be ambient openness. The exact result, retaining the requested order, is ([4], Yes, [[0, 2, 4], [0, 4]], relative_open_need_not_be_ambient_open)."},"skills":["A Relative Open May Need an Ambient Open Extending beyond Y"],"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-20261108-r03-subspace-topology-0047","batch_id":"batch-20261014-advance-20261108-r03-subspace-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1c1d604080b88c8b651fb9646cac712494e311000a0cae379a9ead4bb5bd3f65","family_id":"qf-mod39-subtop-relative-open-audit","seed":202611083047,"semantic_template":"qf-mod39-subtop-relative-open-audit/{\"candidate\":[4],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,4],[0,4],[0]],\"subset\":[1,3,4]}"},"fields":{"title":"A Relative Open May Need an Ambient Open Extending beyond Y","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 2, 3, 4], [0, 2, 4], [0, 4]], and subset Y=[1, 3, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[4] is a duplicate-free subset of Y. Give (candidate, relative_open, parent_open_witnesses, boundary).","answer":{"kind":"expression","value":"([4], Yes, [[0, 2, 4], [0, 4]], relative_open_need_not_be_ambient_open)","explanation":"Check membership in the induced topology, then list every ambient open U with U∩Y equal to the candidate. U need not be contained in Y, and relative openness need not be ambient openness. The exact result, retaining the requested order, is ([4], Yes, [[0, 2, 4], [0, 4]], relative_open_need_not_be_ambient_open)."},"skills":["A Relative Open May Need an Ambient Open Extending beyond Y"],"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-20261108-r03-subspace-topology-0048","batch_id":"batch-20261014-advance-20261108-r03-subspace-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cbbaa816fa9259ea601c17aed61cc89be4f2edce5a1ba06347f79075279813f8","family_id":"qf-mod39-subtop-relative-open-audit","seed":202611083048,"semantic_template":"qf-mod39-subtop-relative-open-audit/{\"candidate\":[4],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,3,4],[0,4],[0]],\"subset\":[1,2,4]}"},"fields":{"title":"A Relative Open May Need an Ambient Open Extending beyond Y","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 3, 4], [0, 3, 4], [0, 4]], and subset Y=[1, 2, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[4] is a duplicate-free subset of Y. Give (candidate, relative_open, parent_open_witnesses, boundary).","answer":{"kind":"expression","value":"([4], Yes, [[0, 3, 4], [0, 4]], relative_open_need_not_be_ambient_open)","explanation":"Check membership in the induced topology, then list every ambient open U with U∩Y equal to the candidate. U need not be contained in Y, and relative openness need not be ambient openness. The exact result, retaining the requested order, is ([4], Yes, [[0, 3, 4], [0, 4]], relative_open_need_not_be_ambient_open)."},"skills":["A Relative Open May Need an Ambient Open Extending beyond Y"],"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-20261108-r03-subspace-topology-0049","batch_id":"batch-20261014-advance-20261108-r03-subspace-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b9d881b5a553e94e52b139e56bb650629889d1634a96377ced69cd427f21fd69","family_id":"qf-mod39-subtop-relative-open-audit","seed":202611083049,"semantic_template":"qf-mod39-subtop-relative-open-audit/{\"candidate\":[4],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,3,4],[0,4],[0]],\"subset\":[1,2,4]}"},"fields":{"title":"A Relative Open May Need an Ambient Open Extending beyond Y","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 2, 3, 4], [0, 3, 4], [0, 4]], and subset Y=[1, 2, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[4] is a duplicate-free subset of Y. Give (candidate, relative_open, parent_open_witnesses, boundary).","answer":{"kind":"expression","value":"([4], Yes, [[0, 3, 4], [0, 4]], relative_open_need_not_be_ambient_open)","explanation":"Check membership in the induced topology, then list every ambient open U with U∩Y equal to the candidate. U need not be contained in Y, and relative openness need not be ambient openness. The exact result, retaining the requested order, is ([4], Yes, [[0, 3, 4], [0, 4]], relative_open_need_not_be_ambient_open)."},"skills":["A Relative Open May Need an Ambient Open Extending beyond Y"],"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-20261108-r03-subspace-topology-0050","batch_id":"batch-20261014-advance-20261108-r03-subspace-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eb03d8056a86ed7e750a150bc6d81a98121207ba1731e12dd7102e08e3e9ffda","family_id":"qf-mod39-subtop-relative-open-audit","seed":202611083050,"semantic_template":"qf-mod39-subtop-relative-open-audit/{\"candidate\":[0],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2],[0,1],[1]],\"subset\":[0,3,4]}"},"fields":{"title":"A Relative Open May Need an Ambient Open Extending beyond Y","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [1]], and subset Y=[0, 3, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[0] is a duplicate-free subset of Y. Give (candidate, relative_open, parent_open_witnesses, boundary).","answer":{"kind":"expression","value":"([0], Yes, [[0, 1], [0, 1, 2]], relative_open_need_not_be_ambient_open)","explanation":"Check membership in the induced topology, then list every ambient open U with U∩Y equal to the candidate. U need not be contained in Y, and relative openness need not be ambient openness. The exact result, retaining the requested order, is ([0], Yes, [[0, 1], [0, 1, 2]], relative_open_need_not_be_ambient_open)."},"skills":["A Relative Open May Need an Ambient Open Extending beyond Y"],"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-20261108-r03-subspace-topology-0051","batch_id":"batch-20261014-advance-20261108-r03-subspace-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cce137e7c48e3c8140bab3f6766ff0261e869d97c5ed4c27678c8656e9762c35","family_id":"qf-mod39-subtop-relative-closed-audit","seed":202611083051,"semantic_template":"qf-mod39-subtop-relative-closed-audit/{\"candidate\":[3,4],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2],[0,1],[0]],\"subset\":[1,3,4]}"},"fields":{"title":"Relative Closedness Uses the Complement inside Y","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]], and subset Y=[1, 3, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[3, 4] is a duplicate-free subset of Y. Give (candidate, relative_complement, relative_closed, boundary).","answer":{"kind":"expression","value":"([3, 4], [1], Yes, complement_taken_inside_Y)","explanation":"Compute Y minus the candidate and check that this relative complement is open in the induced topology. Do not silently replace it by X minus the candidate or infer ambient closedness. The exact result, retaining the requested order, is ([3, 4], [1], Yes, complement_taken_inside_Y)."},"skills":["Relative Closedness Uses the Complement inside Y"],"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-20261108-r03-subspace-topology-0052","batch_id":"batch-20261014-advance-20261108-r03-subspace-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c98e4756b0cba2c20bafdc1e49431ff9e3c0e307bf11de85523bbd7dc08b033b","family_id":"qf-mod39-subtop-relative-closed-audit","seed":202611083052,"semantic_template":"qf-mod39-subtop-relative-closed-audit/{\"candidate\":[3,4],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,2],[0,1],[0]],\"subset\":[1,3,4]}"},"fields":{"title":"Relative Closedness Uses the Complement inside Y","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3, 4], [0, 1, 2, 4]], and subset Y=[1, 3, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[3, 4] is a duplicate-free subset of Y. Give (candidate, relative_complement, relative_closed, boundary).","answer":{"kind":"expression","value":"([3, 4], [1], Yes, complement_taken_inside_Y)","explanation":"Compute Y minus the candidate and check that this relative complement is open in the induced topology. Do not silently replace it by X minus the candidate or infer ambient closedness. The exact result, retaining the requested order, is ([3, 4], [1], Yes, complement_taken_inside_Y)."},"skills":["Relative Closedness Uses the Complement inside Y"],"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-20261108-r03-subspace-topology-0053","batch_id":"batch-20261014-advance-20261108-r03-subspace-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"182ea39a1626e1a0077deacb9403da48bdb3685982504f25c158cdfa29ae51fd","family_id":"qf-mod39-subtop-relative-closed-audit","seed":202611083053,"semantic_template":"qf-mod39-subtop-relative-closed-audit/{\"candidate\":[2,4],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,3],[0,1],[0]],\"subset\":[1,2,4]}"},"fields":{"title":"Relative Closedness Uses the Complement inside Y","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 3]], and subset Y=[1, 2, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[2, 4] is a duplicate-free subset of Y. Give (candidate, relative_complement, relative_closed, boundary).","answer":{"kind":"expression","value":"([2, 4], [1], Yes, complement_taken_inside_Y)","explanation":"Compute Y minus the candidate and check that this relative complement is open in the induced topology. Do not silently replace it by X minus the candidate or infer ambient closedness. The exact result, retaining the requested order, is ([2, 4], [1], Yes, complement_taken_inside_Y)."},"skills":["Relative Closedness Uses the Complement inside Y"],"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-20261108-r03-subspace-topology-0054","batch_id":"batch-20261014-advance-20261108-r03-subspace-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2df78bea57e3b7849f6c91f448bcb2dd3fbb9aa32c3bccc211bfff28e9331686","family_id":"qf-mod39-subtop-relative-closed-audit","seed":202611083054,"semantic_template":"qf-mod39-subtop-relative-closed-audit/{\"candidate\":[2,4],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,3],[0,1],[0]],\"subset\":[1,2,4]}"},"fields":{"title":"Relative Closedness Uses the Complement inside Y","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1], [0, 1, 2, 3, 4], [0, 1, 3], [0, 1, 3, 4]], and subset Y=[1, 2, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[2, 4] is a duplicate-free subset of Y. Give (candidate, relative_complement, relative_closed, boundary).","answer":{"kind":"expression","value":"([2, 4], [1], Yes, complement_taken_inside_Y)","explanation":"Compute Y minus the candidate and check that this relative complement is open in the induced topology. Do not silently replace it by X minus the candidate or infer ambient closedness. The exact result, retaining the requested order, is ([2, 4], [1], Yes, complement_taken_inside_Y)."},"skills":["Relative Closedness Uses the Complement inside Y"],"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-20261108-r03-subspace-topology-0055","batch_id":"batch-20261014-advance-20261108-r03-subspace-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"745f5c8754495fefce4e7b7dacb03e722d4904cf46419700188697f324d88e38","family_id":"qf-mod39-subtop-relative-closed-audit","seed":202611083055,"semantic_template":"qf-mod39-subtop-relative-closed-audit/{\"candidate\":[2,3],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,4],[0,1],[0]],\"subset\":[1,2,3]}"},"fields":{"title":"Relative Closedness Uses the Complement inside Y","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 4]], and subset Y=[1, 2, 3]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[2, 3] is a duplicate-free subset of Y. Give (candidate, relative_complement, relative_closed, boundary).","answer":{"kind":"expression","value":"([2, 3], [1], Yes, complement_taken_inside_Y)","explanation":"Compute Y minus the candidate and check that this relative complement is open in the induced topology. Do not silently replace it by X minus the candidate or infer ambient closedness. The exact result, retaining the requested order, is ([2, 3], [1], Yes, complement_taken_inside_Y)."},"skills":["Relative Closedness Uses the Complement inside Y"],"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-20261108-r03-subspace-topology-0056","batch_id":"batch-20261014-advance-20261108-r03-subspace-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"73c44a009910b3137fac8b34a359e4143136a581135efbf1b285ba99e0677a10","family_id":"qf-mod39-subtop-relative-closed-audit","seed":202611083056,"semantic_template":"qf-mod39-subtop-relative-closed-audit/{\"candidate\":[2,3],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,4],[0,1],[0]],\"subset\":[1,2,3]}"},"fields":{"title":"Relative Closedness Uses the Complement inside Y","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1], [0, 1, 2, 3, 4], [0, 1, 3, 4], [0, 1, 4]], and subset Y=[1, 2, 3]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[2, 3] is a duplicate-free subset of Y. Give (candidate, relative_complement, relative_closed, boundary).","answer":{"kind":"expression","value":"([2, 3], [1], Yes, complement_taken_inside_Y)","explanation":"Compute Y minus the candidate and check that this relative complement is open in the induced topology. Do not silently replace it by X minus the candidate or infer ambient closedness. The exact result, retaining the requested order, is ([2, 3], [1], Yes, complement_taken_inside_Y)."},"skills":["Relative Closedness Uses the Complement inside Y"],"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-20261108-r03-subspace-topology-0057","batch_id":"batch-20261014-advance-20261108-r03-subspace-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"edfe4669e469f64971043fbfe60b8923d1a0879e9fa0f1eb0466eef817a0bbcb","family_id":"qf-mod39-subtop-relative-closed-audit","seed":202611083057,"semantic_template":"qf-mod39-subtop-relative-closed-audit/{\"candidate\":[3,4],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2],[0,2],[0]],\"subset\":[2,3,4]}"},"fields":{"title":"Relative Closedness Uses the Complement inside Y","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 2]], and subset Y=[2, 3, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[3, 4] is a duplicate-free subset of Y. Give (candidate, relative_complement, relative_closed, boundary).","answer":{"kind":"expression","value":"([3, 4], [2], Yes, complement_taken_inside_Y)","explanation":"Compute Y minus the candidate and check that this relative complement is open in the induced topology. Do not silently replace it by X minus the candidate or infer ambient closedness. The exact result, retaining the requested order, is ([3, 4], [2], Yes, complement_taken_inside_Y)."},"skills":["Relative Closedness Uses the Complement inside Y"],"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-20261108-r03-subspace-topology-0058","batch_id":"batch-20261014-advance-20261108-r03-subspace-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4f2461d6342960a2387abb479424d12c5552805b74eb41d84653a673f57eb9ed","family_id":"qf-mod39-subtop-relative-closed-audit","seed":202611083058,"semantic_template":"qf-mod39-subtop-relative-closed-audit/{\"candidate\":[3,4],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,2],[0,2],[0]],\"subset\":[2,3,4]}"},"fields":{"title":"Relative Closedness Uses the Complement inside Y","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 2]], and subset Y=[2, 3, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[3, 4] is a duplicate-free subset of Y. Give (candidate, relative_complement, relative_closed, boundary).","answer":{"kind":"expression","value":"([3, 4], [2], Yes, complement_taken_inside_Y)","explanation":"Compute Y minus the candidate and check that this relative complement is open in the induced topology. Do not silently replace it by X minus the candidate or infer ambient closedness. The exact result, retaining the requested order, is ([3, 4], [2], Yes, complement_taken_inside_Y)."},"skills":["Relative Closedness Uses the Complement inside Y"],"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-20261108-r03-subspace-topology-0059","batch_id":"batch-20261014-advance-20261108-r03-subspace-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7a09d9f8c07c95c17ee927112cae3fcdcb2560b375133e0a197b07667a3582b7","family_id":"qf-mod39-subtop-relative-closed-audit","seed":202611083059,"semantic_template":"qf-mod39-subtop-relative-closed-audit/{\"candidate\":[1,4],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,2,3],[0,2],[0]],\"subset\":[1,2,4]}"},"fields":{"title":"Relative Closedness Uses the Complement inside Y","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 2], [0, 2, 3]], and subset Y=[1, 2, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[1, 4] is a duplicate-free subset of Y. Give (candidate, relative_complement, relative_closed, boundary).","answer":{"kind":"expression","value":"([1, 4], [2], Yes, complement_taken_inside_Y)","explanation":"Compute Y minus the candidate and check that this relative complement is open in the induced topology. Do not silently replace it by X minus the candidate or infer ambient closedness. The exact result, retaining the requested order, is ([1, 4], [2], Yes, complement_taken_inside_Y)."},"skills":["Relative Closedness Uses the Complement inside Y"],"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-20261108-r03-subspace-topology-0060","batch_id":"batch-20261014-advance-20261108-r03-subspace-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"94dcdda3d82eb4881f8603d4d122433060656fb52560462da0de612be764d36c","family_id":"qf-mod39-subtop-relative-closed-audit","seed":202611083060,"semantic_template":"qf-mod39-subtop-relative-closed-audit/{\"candidate\":[1,4],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,3],[0,2],[0]],\"subset\":[1,2,4]}"},"fields":{"title":"Relative Closedness Uses the Complement inside Y","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4]], and subset Y=[1, 2, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[1, 4] is a duplicate-free subset of Y. Give (candidate, relative_complement, relative_closed, boundary).","answer":{"kind":"expression","value":"([1, 4], [2], Yes, complement_taken_inside_Y)","explanation":"Compute Y minus the candidate and check that this relative complement is open in the induced topology. Do not silently replace it by X minus the candidate or infer ambient closedness. The exact result, retaining the requested order, is ([1, 4], [2], Yes, complement_taken_inside_Y)."},"skills":["Relative Closedness Uses the Complement inside Y"],"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-20261108-r03-subspace-topology-0061","batch_id":"batch-20261014-advance-20261108-r03-subspace-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"76adea4cf8ab8b004b1d1a1e0ba88b45d0e1e42d2d0045b06fa1964a3e24bbdf","family_id":"qf-mod39-subtop-relative-closed-audit","seed":202611083061,"semantic_template":"qf-mod39-subtop-relative-closed-audit/{\"candidate\":[1,3],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,2,4],[0,2],[0]],\"subset\":[1,2,3]}"},"fields":{"title":"Relative Closedness Uses the Complement inside Y","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 2], [0, 2, 4]], and subset Y=[1, 2, 3]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[1, 3] is a duplicate-free subset of Y. Give (candidate, relative_complement, relative_closed, boundary).","answer":{"kind":"expression","value":"([1, 3], [2], Yes, complement_taken_inside_Y)","explanation":"Compute Y minus the candidate and check that this relative complement is open in the induced topology. Do not silently replace it by X minus the candidate or infer ambient closedness. The exact result, retaining the requested order, is ([1, 3], [2], Yes, complement_taken_inside_Y)."},"skills":["Relative Closedness Uses the Complement inside Y"],"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-20261108-r03-subspace-topology-0062","batch_id":"batch-20261014-advance-20261108-r03-subspace-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e7bf3b03f5b69e56cbc474c95a3a736d7f53d8bff710a2a290e262df4fcdeb4c","family_id":"qf-mod39-subtop-relative-closed-audit","seed":202611083062,"semantic_template":"qf-mod39-subtop-relative-closed-audit/{\"candidate\":[1,3],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,4],[0,2],[0]],\"subset\":[1,2,3]}"},"fields":{"title":"Relative Closedness Uses the Complement inside Y","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 2], [0, 2, 3, 4], [0, 2, 4]], and subset Y=[1, 2, 3]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[1, 3] is a duplicate-free subset of Y. Give (candidate, relative_complement, relative_closed, boundary).","answer":{"kind":"expression","value":"([1, 3], [2], Yes, complement_taken_inside_Y)","explanation":"Compute Y minus the candidate and check that this relative complement is open in the induced topology. Do not silently replace it by X minus the candidate or infer ambient closedness. The exact result, retaining the requested order, is ([1, 3], [2], Yes, complement_taken_inside_Y)."},"skills":["Relative Closedness Uses the Complement inside Y"],"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-20261108-r03-subspace-topology-0063","batch_id":"batch-20261014-advance-20261108-r03-subspace-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dea1ae35370c957fcdd881fbdae0ef063ba1238767eaffa8e5437773b4dd534a","family_id":"qf-mod39-subtop-relative-closed-audit","seed":202611083063,"semantic_template":"qf-mod39-subtop-relative-closed-audit/{\"candidate\":[2,4],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,3],[0,3],[0]],\"subset\":[2,3,4]}"},"fields":{"title":"Relative Closedness Uses the Complement inside Y","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 3], [0, 3]], and subset Y=[2, 3, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[2, 4] is a duplicate-free subset of Y. Give (candidate, relative_complement, relative_closed, boundary).","answer":{"kind":"expression","value":"([2, 4], [3], Yes, complement_taken_inside_Y)","explanation":"Compute Y minus the candidate and check that this relative complement is open in the induced topology. Do not silently replace it by X minus the candidate or infer ambient closedness. The exact result, retaining the requested order, is ([2, 4], [3], Yes, complement_taken_inside_Y)."},"skills":["Relative Closedness Uses the Complement inside Y"],"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-20261108-r03-subspace-topology-0064","batch_id":"batch-20261014-advance-20261108-r03-subspace-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aa09e7a906139c9e39f64038d07e8f466e166e3320c6fad8c674be74304b2b54","family_id":"qf-mod39-subtop-relative-closed-audit","seed":202611083064,"semantic_template":"qf-mod39-subtop-relative-closed-audit/{\"candidate\":[2,4],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,3],[0,3],[0]],\"subset\":[2,3,4]}"},"fields":{"title":"Relative Closedness Uses the Complement inside Y","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 3], [0, 1, 3, 4], [0, 3]], and subset Y=[2, 3, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[2, 4] is a duplicate-free subset of Y. Give (candidate, relative_complement, relative_closed, boundary).","answer":{"kind":"expression","value":"([2, 4], [3], Yes, complement_taken_inside_Y)","explanation":"Compute Y minus the candidate and check that this relative complement is open in the induced topology. Do not silently replace it by X minus the candidate or infer ambient closedness. The exact result, retaining the requested order, is ([2, 4], [3], Yes, complement_taken_inside_Y)."},"skills":["Relative Closedness Uses the Complement inside Y"],"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-20261108-r03-subspace-topology-0065","batch_id":"batch-20261014-advance-20261108-r03-subspace-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"64612d6de73a078940d601a21a238b07ce3c891f466f86f793e534199f2ae591","family_id":"qf-mod39-subtop-relative-closed-audit","seed":202611083065,"semantic_template":"qf-mod39-subtop-relative-closed-audit/{\"candidate\":[1,4],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,2,3],[0,3],[0]],\"subset\":[1,3,4]}"},"fields":{"title":"Relative Closedness Uses the Complement inside Y","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 2, 3], [0, 3]], and subset Y=[1, 3, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[1, 4] is a duplicate-free subset of Y. Give (candidate, relative_complement, relative_closed, boundary).","answer":{"kind":"expression","value":"([1, 4], [3], Yes, complement_taken_inside_Y)","explanation":"Compute Y minus the candidate and check that this relative complement is open in the induced topology. Do not silently replace it by X minus the candidate or infer ambient closedness. The exact result, retaining the requested order, is ([1, 4], [3], Yes, complement_taken_inside_Y)."},"skills":["Relative Closedness Uses the Complement inside Y"],"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-20261108-r03-subspace-topology-0066","batch_id":"batch-20261014-advance-20261108-r03-subspace-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"32f29fbdc6a3213ba3d380e7f73573f85ed6ba1de28c99d43ea4532f8dc53b93","family_id":"qf-mod39-subtop-relative-closed-audit","seed":202611083066,"semantic_template":"qf-mod39-subtop-relative-closed-audit/{\"candidate\":[1,4],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,3],[0,3],[0]],\"subset\":[1,3,4]}"},"fields":{"title":"Relative Closedness Uses the Complement inside Y","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 2, 3], [0, 2, 3, 4], [0, 3]], and subset Y=[1, 3, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[1, 4] is a duplicate-free subset of Y. Give (candidate, relative_complement, relative_closed, boundary).","answer":{"kind":"expression","value":"([1, 4], [3], Yes, complement_taken_inside_Y)","explanation":"Compute Y minus the candidate and check that this relative complement is open in the induced topology. Do not silently replace it by X minus the candidate or infer ambient closedness. The exact result, retaining the requested order, is ([1, 4], [3], Yes, complement_taken_inside_Y)."},"skills":["Relative Closedness Uses the Complement inside Y"],"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-20261108-r03-subspace-topology-0067","batch_id":"batch-20261014-advance-20261108-r03-subspace-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9faad30131a2683f25376416f1c626494563b735436e5cc7edb0c8ad0cee491e","family_id":"qf-mod39-subtop-relative-closed-audit","seed":202611083067,"semantic_template":"qf-mod39-subtop-relative-closed-audit/{\"candidate\":[1,2],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,3,4],[0,3],[0]],\"subset\":[1,2,3]}"},"fields":{"title":"Relative Closedness Uses the Complement inside Y","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 3, 4], [0, 3], [0, 3, 4]], and subset Y=[1, 2, 3]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[1, 2] is a duplicate-free subset of Y. Give (candidate, relative_complement, relative_closed, boundary).","answer":{"kind":"expression","value":"([1, 2], [3], Yes, complement_taken_inside_Y)","explanation":"Compute Y minus the candidate and check that this relative complement is open in the induced topology. Do not silently replace it by X minus the candidate or infer ambient closedness. The exact result, retaining the requested order, is ([1, 2], [3], Yes, complement_taken_inside_Y)."},"skills":["Relative Closedness Uses the Complement inside Y"],"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-20261108-r03-subspace-topology-0068","batch_id":"batch-20261014-advance-20261108-r03-subspace-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"da5f93c344e08bca67a6af02d962716349b8f85eb9dae4baf8f427eab676ed63","family_id":"qf-mod39-subtop-relative-closed-audit","seed":202611083068,"semantic_template":"qf-mod39-subtop-relative-closed-audit/{\"candidate\":[1,2],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,3,4],[0,3],[0]],\"subset\":[1,2,3]}"},"fields":{"title":"Relative Closedness Uses the Complement inside Y","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 2, 3, 4], [0, 3], [0, 3, 4]], and subset Y=[1, 2, 3]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[1, 2] is a duplicate-free subset of Y. Give (candidate, relative_complement, relative_closed, boundary).","answer":{"kind":"expression","value":"([1, 2], [3], Yes, complement_taken_inside_Y)","explanation":"Compute Y minus the candidate and check that this relative complement is open in the induced topology. Do not silently replace it by X minus the candidate or infer ambient closedness. The exact result, retaining the requested order, is ([1, 2], [3], Yes, complement_taken_inside_Y)."},"skills":["Relative Closedness Uses the Complement inside Y"],"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-20261108-r03-subspace-topology-0069","batch_id":"batch-20261014-advance-20261108-r03-subspace-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f8a99c4041c0431735c0a1ee6148b2dc583f67f559c6d71e6bd1b673f8ad6af3","family_id":"qf-mod39-subtop-relative-closed-audit","seed":202611083069,"semantic_template":"qf-mod39-subtop-relative-closed-audit/{\"candidate\":[2,3],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,4],[0,4],[0]],\"subset\":[2,3,4]}"},"fields":{"title":"Relative Closedness Uses the Complement inside Y","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 4], [0, 4]], and subset Y=[2, 3, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[2, 3] is a duplicate-free subset of Y. Give (candidate, relative_complement, relative_closed, boundary).","answer":{"kind":"expression","value":"([2, 3], [4], Yes, complement_taken_inside_Y)","explanation":"Compute Y minus the candidate and check that this relative complement is open in the induced topology. Do not silently replace it by X minus the candidate or infer ambient closedness. The exact result, retaining the requested order, is ([2, 3], [4], Yes, complement_taken_inside_Y)."},"skills":["Relative Closedness Uses the Complement inside Y"],"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-20261108-r03-subspace-topology-0070","batch_id":"batch-20261014-advance-20261108-r03-subspace-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"899fa2ee07f76023b88d37fedcfd444e79b345c926a0932aa8dca69333630bec","family_id":"qf-mod39-subtop-relative-closed-audit","seed":202611083070,"semantic_template":"qf-mod39-subtop-relative-closed-audit/{\"candidate\":[2,3],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,4],[0,4],[0]],\"subset\":[2,3,4]}"},"fields":{"title":"Relative Closedness Uses the Complement inside Y","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 3, 4], [0, 1, 4], [0, 4]], and subset Y=[2, 3, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[2, 3] is a duplicate-free subset of Y. Give (candidate, relative_complement, relative_closed, boundary).","answer":{"kind":"expression","value":"([2, 3], [4], Yes, complement_taken_inside_Y)","explanation":"Compute Y minus the candidate and check that this relative complement is open in the induced topology. Do not silently replace it by X minus the candidate or infer ambient closedness. The exact result, retaining the requested order, is ([2, 3], [4], Yes, complement_taken_inside_Y)."},"skills":["Relative Closedness Uses the Complement inside Y"],"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-20261108-r03-subspace-topology-0071","batch_id":"batch-20261014-advance-20261108-r03-subspace-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b2e726732e90996ef53515cccb8e30b8841fdcbfd9c043368a3511e5b63a697a","family_id":"qf-mod39-subtop-relative-closed-audit","seed":202611083071,"semantic_template":"qf-mod39-subtop-relative-closed-audit/{\"candidate\":[1,3],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,2,4],[0,4],[0]],\"subset\":[1,3,4]}"},"fields":{"title":"Relative Closedness Uses the Complement inside Y","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 2, 4], [0, 4]], and subset Y=[1, 3, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[1, 3] is a duplicate-free subset of Y. Give (candidate, relative_complement, relative_closed, boundary).","answer":{"kind":"expression","value":"([1, 3], [4], Yes, complement_taken_inside_Y)","explanation":"Compute Y minus the candidate and check that this relative complement is open in the induced topology. Do not silently replace it by X minus the candidate or infer ambient closedness. The exact result, retaining the requested order, is ([1, 3], [4], Yes, complement_taken_inside_Y)."},"skills":["Relative Closedness Uses the Complement inside Y"],"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-20261108-r03-subspace-topology-0072","batch_id":"batch-20261014-advance-20261108-r03-subspace-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"63d1bf816c28b9f942d9453d59f026f3ed9a02d400856db6776cfb99483dff0a","family_id":"qf-mod39-subtop-relative-closed-audit","seed":202611083072,"semantic_template":"qf-mod39-subtop-relative-closed-audit/{\"candidate\":[1,3],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,4],[0,4],[0]],\"subset\":[1,3,4]}"},"fields":{"title":"Relative Closedness Uses the Complement inside Y","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 2, 3, 4], [0, 2, 4], [0, 4]], and subset Y=[1, 3, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[1, 3] is a duplicate-free subset of Y. Give (candidate, relative_complement, relative_closed, boundary).","answer":{"kind":"expression","value":"([1, 3], [4], Yes, complement_taken_inside_Y)","explanation":"Compute Y minus the candidate and check that this relative complement is open in the induced topology. Do not silently replace it by X minus the candidate or infer ambient closedness. The exact result, retaining the requested order, is ([1, 3], [4], Yes, complement_taken_inside_Y)."},"skills":["Relative Closedness Uses the Complement inside Y"],"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-20261108-r03-subspace-topology-0073","batch_id":"batch-20261014-advance-20261108-r03-subspace-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6245032945f4f55e9c4effa592472b3262366d7101112a4fb2640f2c6fca4e61","family_id":"qf-mod39-subtop-relative-closed-audit","seed":202611083073,"semantic_template":"qf-mod39-subtop-relative-closed-audit/{\"candidate\":[1,2],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,3,4],[0,4],[0]],\"subset\":[1,2,4]}"},"fields":{"title":"Relative Closedness Uses the Complement inside Y","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 3, 4], [0, 3, 4], [0, 4]], and subset Y=[1, 2, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[1, 2] is a duplicate-free subset of Y. Give (candidate, relative_complement, relative_closed, boundary).","answer":{"kind":"expression","value":"([1, 2], [4], Yes, complement_taken_inside_Y)","explanation":"Compute Y minus the candidate and check that this relative complement is open in the induced topology. Do not silently replace it by X minus the candidate or infer ambient closedness. The exact result, retaining the requested order, is ([1, 2], [4], Yes, complement_taken_inside_Y)."},"skills":["Relative Closedness Uses the Complement inside Y"],"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-20261108-r03-subspace-topology-0074","batch_id":"batch-20261014-advance-20261108-r03-subspace-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c1e4b22a8b04982a7aa2aaa4daf45d39f8c4189c0d0a11d91491db8beaeec79d","family_id":"qf-mod39-subtop-relative-closed-audit","seed":202611083074,"semantic_template":"qf-mod39-subtop-relative-closed-audit/{\"candidate\":[1,2],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,3,4],[0,4],[0]],\"subset\":[1,2,4]}"},"fields":{"title":"Relative Closedness Uses the Complement inside Y","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 2, 3, 4], [0, 3, 4], [0, 4]], and subset Y=[1, 2, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[1, 2] is a duplicate-free subset of Y. Give (candidate, relative_complement, relative_closed, boundary).","answer":{"kind":"expression","value":"([1, 2], [4], Yes, complement_taken_inside_Y)","explanation":"Compute Y minus the candidate and check that this relative complement is open in the induced topology. Do not silently replace it by X minus the candidate or infer ambient closedness. The exact result, retaining the requested order, is ([1, 2], [4], Yes, complement_taken_inside_Y)."},"skills":["Relative Closedness Uses the Complement inside Y"],"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-20261108-r03-subspace-topology-0075","batch_id":"batch-20261014-advance-20261108-r03-subspace-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2487b367d9895a9aa20c4eaf822a7fada250e3171ea1c9862dc80282347ea322","family_id":"qf-mod39-subtop-relative-closed-audit","seed":202611083075,"semantic_template":"qf-mod39-subtop-relative-closed-audit/{\"candidate\":[3,4],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2],[0,1],[1]],\"subset\":[0,3,4]}"},"fields":{"title":"Relative Closedness Uses the Complement inside Y","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [1]], and subset Y=[0, 3, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[3, 4] is a duplicate-free subset of Y. Give (candidate, relative_complement, relative_closed, boundary).","answer":{"kind":"expression","value":"([3, 4], [0], Yes, complement_taken_inside_Y)","explanation":"Compute Y minus the candidate and check that this relative complement is open in the induced topology. Do not silently replace it by X minus the candidate or infer ambient closedness. The exact result, retaining the requested order, is ([3, 4], [0], Yes, complement_taken_inside_Y)."},"skills":["Relative Closedness Uses the Complement inside Y"],"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-20261108-r03-subspace-topology-0076","batch_id":"batch-20261014-advance-20261108-r03-subspace-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0014d9a5964f453795177d932deb6204438b3441b11a453607abb57acd8b9f30","family_id":"qf-mod39-subtop-ambient-boundary","seed":202611083076,"semantic_template":"qf-mod39-subtop-ambient-boundary/{\"candidate\":[1,3,4],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2],[0,1],[0]],\"subset\":[1,3,4]}"},"fields":{"title":"Relative-Open but Not Ambient-Open Is a Valid Boundary Case","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]], and subset Y=[1, 3, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[1, 3, 4] is a duplicate-free subset of Y. Give (candidate, relative_open, ambient_open, intersection_certificate, boundary).","answer":{"kind":"expression","value":"([1, 3, 4], Yes, No, Yes, O_intersect_Y_not_O_subset_Y)","explanation":"Require the actual candidate to be relatively open but not ambient open. The parent-open intersection certificate does not require its ambient witness U to lie inside Y. The exact result, retaining the requested order, is ([1, 3, 4], Yes, No, Yes, O_intersect_Y_not_O_subset_Y)."},"skills":["Relative-Open but Not Ambient-Open Is a Valid Boundary Case"],"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-20261108-r03-subspace-topology-0077","batch_id":"batch-20261014-advance-20261108-r03-subspace-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"98c7606d87aff2cd4564f0ec9d833d756977c4e259012f9aa2b19ba7423c05b8","family_id":"qf-mod39-subtop-ambient-boundary","seed":202611083077,"semantic_template":"qf-mod39-subtop-ambient-boundary/{\"candidate\":[1,3,4],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,2],[0,1],[0]],\"subset\":[1,3,4]}"},"fields":{"title":"Relative-Open but Not Ambient-Open Is a Valid Boundary Case","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3, 4], [0, 1, 2, 4]], and subset Y=[1, 3, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[1, 3, 4] is a duplicate-free subset of Y. Give (candidate, relative_open, ambient_open, intersection_certificate, boundary).","answer":{"kind":"expression","value":"([1, 3, 4], Yes, No, Yes, O_intersect_Y_not_O_subset_Y)","explanation":"Require the actual candidate to be relatively open but not ambient open. The parent-open intersection certificate does not require its ambient witness U to lie inside Y. The exact result, retaining the requested order, is ([1, 3, 4], Yes, No, Yes, O_intersect_Y_not_O_subset_Y)."},"skills":["Relative-Open but Not Ambient-Open Is a Valid Boundary Case"],"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-20261108-r03-subspace-topology-0078","batch_id":"batch-20261014-advance-20261108-r03-subspace-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e9d29dc15f99f6310a1cf5063bb636512af5b27f01638ec0adc2b3f910e9d66c","family_id":"qf-mod39-subtop-ambient-boundary","seed":202611083078,"semantic_template":"qf-mod39-subtop-ambient-boundary/{\"candidate\":[1,2,4],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,3],[0,1],[0]],\"subset\":[1,2,4]}"},"fields":{"title":"Relative-Open but Not Ambient-Open Is a Valid Boundary Case","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 3]], and subset Y=[1, 2, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[1, 2, 4] is a duplicate-free subset of Y. Give (candidate, relative_open, ambient_open, intersection_certificate, boundary).","answer":{"kind":"expression","value":"([1, 2, 4], Yes, No, Yes, O_intersect_Y_not_O_subset_Y)","explanation":"Require the actual candidate to be relatively open but not ambient open. The parent-open intersection certificate does not require its ambient witness U to lie inside Y. The exact result, retaining the requested order, is ([1, 2, 4], Yes, No, Yes, O_intersect_Y_not_O_subset_Y)."},"skills":["Relative-Open but Not Ambient-Open Is a Valid Boundary Case"],"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-20261108-r03-subspace-topology-0079","batch_id":"batch-20261014-advance-20261108-r03-subspace-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1f4f77fd831f46b380df8011f59cb26fc454725cf21c5d757fe5df054d730628","family_id":"qf-mod39-subtop-ambient-boundary","seed":202611083079,"semantic_template":"qf-mod39-subtop-ambient-boundary/{\"candidate\":[1,2,4],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,3],[0,1],[0]],\"subset\":[1,2,4]}"},"fields":{"title":"Relative-Open but Not Ambient-Open Is a Valid Boundary Case","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1], [0, 1, 2, 3, 4], [0, 1, 3], [0, 1, 3, 4]], and subset Y=[1, 2, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[1, 2, 4] is a duplicate-free subset of Y. Give (candidate, relative_open, ambient_open, intersection_certificate, boundary).","answer":{"kind":"expression","value":"([1, 2, 4], Yes, No, Yes, O_intersect_Y_not_O_subset_Y)","explanation":"Require the actual candidate to be relatively open but not ambient open. The parent-open intersection certificate does not require its ambient witness U to lie inside Y. The exact result, retaining the requested order, is ([1, 2, 4], Yes, No, Yes, O_intersect_Y_not_O_subset_Y)."},"skills":["Relative-Open but Not Ambient-Open Is a Valid Boundary Case"],"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-20261108-r03-subspace-topology-0080","batch_id":"batch-20261014-advance-20261108-r03-subspace-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bbd6f8718971c91c3d64f387fe1e1897b614f7c6ae12aebf014e0b9aa18078fb","family_id":"qf-mod39-subtop-ambient-boundary","seed":202611083080,"semantic_template":"qf-mod39-subtop-ambient-boundary/{\"candidate\":[1,2,3],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,4],[0,1],[0]],\"subset\":[1,2,3]}"},"fields":{"title":"Relative-Open but Not Ambient-Open Is a Valid Boundary Case","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 4]], and subset Y=[1, 2, 3]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[1, 2, 3] is a duplicate-free subset of Y. Give (candidate, relative_open, ambient_open, intersection_certificate, boundary).","answer":{"kind":"expression","value":"([1, 2, 3], Yes, No, Yes, O_intersect_Y_not_O_subset_Y)","explanation":"Require the actual candidate to be relatively open but not ambient open. The parent-open intersection certificate does not require its ambient witness U to lie inside Y. The exact result, retaining the requested order, is ([1, 2, 3], Yes, No, Yes, O_intersect_Y_not_O_subset_Y)."},"skills":["Relative-Open but Not Ambient-Open Is a Valid Boundary Case"],"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-20261108-r03-subspace-topology-0081","batch_id":"batch-20261014-advance-20261108-r03-subspace-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9978c73fe66e08a1107e819e779e73143eb8fb6c769b93d5d0e036afe6932b93","family_id":"qf-mod39-subtop-ambient-boundary","seed":202611083081,"semantic_template":"qf-mod39-subtop-ambient-boundary/{\"candidate\":[1,2,3],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,4],[0,1],[0]],\"subset\":[1,2,3]}"},"fields":{"title":"Relative-Open but Not Ambient-Open Is a Valid Boundary Case","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1], [0, 1, 2, 3, 4], [0, 1, 3, 4], [0, 1, 4]], and subset Y=[1, 2, 3]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[1, 2, 3] is a duplicate-free subset of Y. Give (candidate, relative_open, ambient_open, intersection_certificate, boundary).","answer":{"kind":"expression","value":"([1, 2, 3], Yes, No, Yes, O_intersect_Y_not_O_subset_Y)","explanation":"Require the actual candidate to be relatively open but not ambient open. The parent-open intersection certificate does not require its ambient witness U to lie inside Y. The exact result, retaining the requested order, is ([1, 2, 3], Yes, No, Yes, O_intersect_Y_not_O_subset_Y)."},"skills":["Relative-Open but Not Ambient-Open Is a Valid Boundary Case"],"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-20261108-r03-subspace-topology-0082","batch_id":"batch-20261014-advance-20261108-r03-subspace-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5b464aa1a01d552d879a7eaa013d5264bb0e52e4d9a19adfaa6cc6e78fb9dfa7","family_id":"qf-mod39-subtop-ambient-boundary","seed":202611083082,"semantic_template":"qf-mod39-subtop-ambient-boundary/{\"candidate\":[2,3,4],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2],[0,2],[0]],\"subset\":[2,3,4]}"},"fields":{"title":"Relative-Open but Not Ambient-Open Is a Valid Boundary Case","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 2]], and subset Y=[2, 3, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[2, 3, 4] is a duplicate-free subset of Y. Give (candidate, relative_open, ambient_open, intersection_certificate, boundary).","answer":{"kind":"expression","value":"([2, 3, 4], Yes, No, Yes, O_intersect_Y_not_O_subset_Y)","explanation":"Require the actual candidate to be relatively open but not ambient open. The parent-open intersection certificate does not require its ambient witness U to lie inside Y. The exact result, retaining the requested order, is ([2, 3, 4], Yes, No, Yes, O_intersect_Y_not_O_subset_Y)."},"skills":["Relative-Open but Not Ambient-Open Is a Valid Boundary Case"],"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-20261108-r03-subspace-topology-0083","batch_id":"batch-20261014-advance-20261108-r03-subspace-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c5b515ab6fb57cd31c9195067e3b0097dd6740c2cab089d4064fa31fa9fed72f","family_id":"qf-mod39-subtop-ambient-boundary","seed":202611083083,"semantic_template":"qf-mod39-subtop-ambient-boundary/{\"candidate\":[2,3,4],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,2],[0,2],[0]],\"subset\":[2,3,4]}"},"fields":{"title":"Relative-Open but Not Ambient-Open Is a Valid Boundary Case","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 2]], and subset Y=[2, 3, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[2, 3, 4] is a duplicate-free subset of Y. Give (candidate, relative_open, ambient_open, intersection_certificate, boundary).","answer":{"kind":"expression","value":"([2, 3, 4], Yes, No, Yes, O_intersect_Y_not_O_subset_Y)","explanation":"Require the actual candidate to be relatively open but not ambient open. The parent-open intersection certificate does not require its ambient witness U to lie inside Y. The exact result, retaining the requested order, is ([2, 3, 4], Yes, No, Yes, O_intersect_Y_not_O_subset_Y)."},"skills":["Relative-Open but Not Ambient-Open Is a Valid Boundary Case"],"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-20261108-r03-subspace-topology-0084","batch_id":"batch-20261014-advance-20261108-r03-subspace-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0f7e9ca84cb3fa782dfd143611f145be3df69b133bc147fff2654b5a48c84676","family_id":"qf-mod39-subtop-ambient-boundary","seed":202611083084,"semantic_template":"qf-mod39-subtop-ambient-boundary/{\"candidate\":[1,2,4],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,2,3],[0,2],[0]],\"subset\":[1,2,4]}"},"fields":{"title":"Relative-Open but Not Ambient-Open Is a Valid Boundary Case","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 2], [0, 2, 3]], and subset Y=[1, 2, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[1, 2, 4] is a duplicate-free subset of Y. Give (candidate, relative_open, ambient_open, intersection_certificate, boundary).","answer":{"kind":"expression","value":"([1, 2, 4], Yes, No, Yes, O_intersect_Y_not_O_subset_Y)","explanation":"Require the actual candidate to be relatively open but not ambient open. The parent-open intersection certificate does not require its ambient witness U to lie inside Y. The exact result, retaining the requested order, is ([1, 2, 4], Yes, No, Yes, O_intersect_Y_not_O_subset_Y)."},"skills":["Relative-Open but Not Ambient-Open Is a Valid Boundary Case"],"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-20261108-r03-subspace-topology-0085","batch_id":"batch-20261014-advance-20261108-r03-subspace-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2b1ea90bb6620a42d2a6639f3cb3fda40290cb881cc6c9cf9eff72bc49c98f3b","family_id":"qf-mod39-subtop-ambient-boundary","seed":202611083085,"semantic_template":"qf-mod39-subtop-ambient-boundary/{\"candidate\":[1,2,4],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,3],[0,2],[0]],\"subset\":[1,2,4]}"},"fields":{"title":"Relative-Open but Not Ambient-Open Is a Valid Boundary Case","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4]], and subset Y=[1, 2, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[1, 2, 4] is a duplicate-free subset of Y. Give (candidate, relative_open, ambient_open, intersection_certificate, boundary).","answer":{"kind":"expression","value":"([1, 2, 4], Yes, No, Yes, O_intersect_Y_not_O_subset_Y)","explanation":"Require the actual candidate to be relatively open but not ambient open. The parent-open intersection certificate does not require its ambient witness U to lie inside Y. The exact result, retaining the requested order, is ([1, 2, 4], Yes, No, Yes, O_intersect_Y_not_O_subset_Y)."},"skills":["Relative-Open but Not Ambient-Open Is a Valid Boundary Case"],"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-20261108-r03-subspace-topology-0086","batch_id":"batch-20261014-advance-20261108-r03-subspace-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a8556a1f655bf7c238c3ba5f7399e0bb5f7c645239b4978d2efde09404abe1de","family_id":"qf-mod39-subtop-ambient-boundary","seed":202611083086,"semantic_template":"qf-mod39-subtop-ambient-boundary/{\"candidate\":[1,2,3],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,2,4],[0,2],[0]],\"subset\":[1,2,3]}"},"fields":{"title":"Relative-Open but Not Ambient-Open Is a Valid Boundary Case","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 2], [0, 2, 4]], and subset Y=[1, 2, 3]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[1, 2, 3] is a duplicate-free subset of Y. Give (candidate, relative_open, ambient_open, intersection_certificate, boundary).","answer":{"kind":"expression","value":"([1, 2, 3], Yes, No, Yes, O_intersect_Y_not_O_subset_Y)","explanation":"Require the actual candidate to be relatively open but not ambient open. The parent-open intersection certificate does not require its ambient witness U to lie inside Y. The exact result, retaining the requested order, is ([1, 2, 3], Yes, No, Yes, O_intersect_Y_not_O_subset_Y)."},"skills":["Relative-Open but Not Ambient-Open Is a Valid Boundary Case"],"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-20261108-r03-subspace-topology-0087","batch_id":"batch-20261014-advance-20261108-r03-subspace-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bc0c709180398c8c8fb50e33de4766003c92f9b09fef5ed37ba951dbcd1412cc","family_id":"qf-mod39-subtop-ambient-boundary","seed":202611083087,"semantic_template":"qf-mod39-subtop-ambient-boundary/{\"candidate\":[1,2,3],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,4],[0,2],[0]],\"subset\":[1,2,3]}"},"fields":{"title":"Relative-Open but Not Ambient-Open Is a Valid Boundary Case","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 2], [0, 2, 3, 4], [0, 2, 4]], and subset Y=[1, 2, 3]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[1, 2, 3] is a duplicate-free subset of Y. Give (candidate, relative_open, ambient_open, intersection_certificate, boundary).","answer":{"kind":"expression","value":"([1, 2, 3], Yes, No, Yes, O_intersect_Y_not_O_subset_Y)","explanation":"Require the actual candidate to be relatively open but not ambient open. The parent-open intersection certificate does not require its ambient witness U to lie inside Y. The exact result, retaining the requested order, is ([1, 2, 3], Yes, No, Yes, O_intersect_Y_not_O_subset_Y)."},"skills":["Relative-Open but Not Ambient-Open Is a Valid Boundary Case"],"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-20261108-r03-subspace-topology-0088","batch_id":"batch-20261014-advance-20261108-r03-subspace-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"98edbcb5a1817d2fce7401d67d52ceb11574567a5c79e5eb5a820ada2fa27c50","family_id":"qf-mod39-subtop-ambient-boundary","seed":202611083088,"semantic_template":"qf-mod39-subtop-ambient-boundary/{\"candidate\":[2,3,4],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,3],[0,3],[0]],\"subset\":[2,3,4]}"},"fields":{"title":"Relative-Open but Not Ambient-Open Is a Valid Boundary Case","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 3], [0, 3]], and subset Y=[2, 3, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[2, 3, 4] is a duplicate-free subset of Y. Give (candidate, relative_open, ambient_open, intersection_certificate, boundary).","answer":{"kind":"expression","value":"([2, 3, 4], Yes, No, Yes, O_intersect_Y_not_O_subset_Y)","explanation":"Require the actual candidate to be relatively open but not ambient open. The parent-open intersection certificate does not require its ambient witness U to lie inside Y. The exact result, retaining the requested order, is ([2, 3, 4], Yes, No, Yes, O_intersect_Y_not_O_subset_Y)."},"skills":["Relative-Open but Not Ambient-Open Is a Valid Boundary Case"],"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-20261108-r03-subspace-topology-0089","batch_id":"batch-20261014-advance-20261108-r03-subspace-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4a665b3ed494261890fb24c868a121db7d85b5fb0de1703397bbf194a3b87925","family_id":"qf-mod39-subtop-ambient-boundary","seed":202611083089,"semantic_template":"qf-mod39-subtop-ambient-boundary/{\"candidate\":[2,3,4],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,3],[0,3],[0]],\"subset\":[2,3,4]}"},"fields":{"title":"Relative-Open but Not Ambient-Open Is a Valid Boundary Case","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 3], [0, 1, 3, 4], [0, 3]], and subset Y=[2, 3, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[2, 3, 4] is a duplicate-free subset of Y. Give (candidate, relative_open, ambient_open, intersection_certificate, boundary).","answer":{"kind":"expression","value":"([2, 3, 4], Yes, No, Yes, O_intersect_Y_not_O_subset_Y)","explanation":"Require the actual candidate to be relatively open but not ambient open. The parent-open intersection certificate does not require its ambient witness U to lie inside Y. The exact result, retaining the requested order, is ([2, 3, 4], Yes, No, Yes, O_intersect_Y_not_O_subset_Y)."},"skills":["Relative-Open but Not Ambient-Open Is a Valid Boundary Case"],"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-20261108-r03-subspace-topology-0090","batch_id":"batch-20261014-advance-20261108-r03-subspace-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b5f6453222a82d1c85bc9741714bfcc88425b3473b583a1929d7db6f34af741e","family_id":"qf-mod39-subtop-ambient-boundary","seed":202611083090,"semantic_template":"qf-mod39-subtop-ambient-boundary/{\"candidate\":[1,3,4],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,2,3],[0,3],[0]],\"subset\":[1,3,4]}"},"fields":{"title":"Relative-Open but Not Ambient-Open Is a Valid Boundary Case","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 2, 3], [0, 3]], and subset Y=[1, 3, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[1, 3, 4] is a duplicate-free subset of Y. Give (candidate, relative_open, ambient_open, intersection_certificate, boundary).","answer":{"kind":"expression","value":"([1, 3, 4], Yes, No, Yes, O_intersect_Y_not_O_subset_Y)","explanation":"Require the actual candidate to be relatively open but not ambient open. The parent-open intersection certificate does not require its ambient witness U to lie inside Y. The exact result, retaining the requested order, is ([1, 3, 4], Yes, No, Yes, O_intersect_Y_not_O_subset_Y)."},"skills":["Relative-Open but Not Ambient-Open Is a Valid Boundary Case"],"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-20261108-r03-subspace-topology-0091","batch_id":"batch-20261014-advance-20261108-r03-subspace-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"791eb49cfbe58e7552ca66546e35f7b5d5d8a3b9aff4f4d5f98433962f44382b","family_id":"qf-mod39-subtop-ambient-boundary","seed":202611083091,"semantic_template":"qf-mod39-subtop-ambient-boundary/{\"candidate\":[1,3,4],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,3],[0,3],[0]],\"subset\":[1,3,4]}"},"fields":{"title":"Relative-Open but Not Ambient-Open Is a Valid Boundary Case","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 2, 3], [0, 2, 3, 4], [0, 3]], and subset Y=[1, 3, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[1, 3, 4] is a duplicate-free subset of Y. Give (candidate, relative_open, ambient_open, intersection_certificate, boundary).","answer":{"kind":"expression","value":"([1, 3, 4], Yes, No, Yes, O_intersect_Y_not_O_subset_Y)","explanation":"Require the actual candidate to be relatively open but not ambient open. The parent-open intersection certificate does not require its ambient witness U to lie inside Y. The exact result, retaining the requested order, is ([1, 3, 4], Yes, No, Yes, O_intersect_Y_not_O_subset_Y)."},"skills":["Relative-Open but Not Ambient-Open Is a Valid Boundary Case"],"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-20261108-r03-subspace-topology-0092","batch_id":"batch-20261014-advance-20261108-r03-subspace-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a6e24585f3edb4e6f2e065954fdabd7a084398dc29a8c751a3da3d4f9fbc87a6","family_id":"qf-mod39-subtop-ambient-boundary","seed":202611083092,"semantic_template":"qf-mod39-subtop-ambient-boundary/{\"candidate\":[1,2,3],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,3,4],[0,3],[0]],\"subset\":[1,2,3]}"},"fields":{"title":"Relative-Open but Not Ambient-Open Is a Valid Boundary Case","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 3, 4], [0, 3], [0, 3, 4]], and subset Y=[1, 2, 3]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[1, 2, 3] is a duplicate-free subset of Y. Give (candidate, relative_open, ambient_open, intersection_certificate, boundary).","answer":{"kind":"expression","value":"([1, 2, 3], Yes, No, Yes, O_intersect_Y_not_O_subset_Y)","explanation":"Require the actual candidate to be relatively open but not ambient open. The parent-open intersection certificate does not require its ambient witness U to lie inside Y. The exact result, retaining the requested order, is ([1, 2, 3], Yes, No, Yes, O_intersect_Y_not_O_subset_Y)."},"skills":["Relative-Open but Not Ambient-Open Is a Valid Boundary Case"],"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-20261108-r03-subspace-topology-0093","batch_id":"batch-20261014-advance-20261108-r03-subspace-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0087f098fb7da3a3f4424b9b0c342b28e46a4d4e6a85fcddf85338e063ec79a0","family_id":"qf-mod39-subtop-ambient-boundary","seed":202611083093,"semantic_template":"qf-mod39-subtop-ambient-boundary/{\"candidate\":[1,2,3],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,3,4],[0,3],[0]],\"subset\":[1,2,3]}"},"fields":{"title":"Relative-Open but Not Ambient-Open Is a Valid Boundary Case","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 2, 3, 4], [0, 3], [0, 3, 4]], and subset Y=[1, 2, 3]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[1, 2, 3] is a duplicate-free subset of Y. Give (candidate, relative_open, ambient_open, intersection_certificate, boundary).","answer":{"kind":"expression","value":"([1, 2, 3], Yes, No, Yes, O_intersect_Y_not_O_subset_Y)","explanation":"Require the actual candidate to be relatively open but not ambient open. The parent-open intersection certificate does not require its ambient witness U to lie inside Y. The exact result, retaining the requested order, is ([1, 2, 3], Yes, No, Yes, O_intersect_Y_not_O_subset_Y)."},"skills":["Relative-Open but Not Ambient-Open Is a Valid Boundary Case"],"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-20261108-r03-subspace-topology-0094","batch_id":"batch-20261014-advance-20261108-r03-subspace-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8684d169d735fd09967189ec1dbd6ddc84d938fc8c5429dcff3b66af021c7b66","family_id":"qf-mod39-subtop-ambient-boundary","seed":202611083094,"semantic_template":"qf-mod39-subtop-ambient-boundary/{\"candidate\":[2,3,4],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,4],[0,4],[0]],\"subset\":[2,3,4]}"},"fields":{"title":"Relative-Open but Not Ambient-Open Is a Valid Boundary Case","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 4], [0, 4]], and subset Y=[2, 3, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[2, 3, 4] is a duplicate-free subset of Y. Give (candidate, relative_open, ambient_open, intersection_certificate, boundary).","answer":{"kind":"expression","value":"([2, 3, 4], Yes, No, Yes, O_intersect_Y_not_O_subset_Y)","explanation":"Require the actual candidate to be relatively open but not ambient open. The parent-open intersection certificate does not require its ambient witness U to lie inside Y. The exact result, retaining the requested order, is ([2, 3, 4], Yes, No, Yes, O_intersect_Y_not_O_subset_Y)."},"skills":["Relative-Open but Not Ambient-Open Is a Valid Boundary Case"],"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-20261108-r03-subspace-topology-0095","batch_id":"batch-20261014-advance-20261108-r03-subspace-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b328feb021121c2fbca150c40bba31de1f2a40d870f2548ce4650d16cb27a43b","family_id":"qf-mod39-subtop-ambient-boundary","seed":202611083095,"semantic_template":"qf-mod39-subtop-ambient-boundary/{\"candidate\":[2,3,4],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,4],[0,4],[0]],\"subset\":[2,3,4]}"},"fields":{"title":"Relative-Open but Not Ambient-Open Is a Valid Boundary Case","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 3, 4], [0, 1, 4], [0, 4]], and subset Y=[2, 3, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[2, 3, 4] is a duplicate-free subset of Y. Give (candidate, relative_open, ambient_open, intersection_certificate, boundary).","answer":{"kind":"expression","value":"([2, 3, 4], Yes, No, Yes, O_intersect_Y_not_O_subset_Y)","explanation":"Require the actual candidate to be relatively open but not ambient open. The parent-open intersection certificate does not require its ambient witness U to lie inside Y. The exact result, retaining the requested order, is ([2, 3, 4], Yes, No, Yes, O_intersect_Y_not_O_subset_Y)."},"skills":["Relative-Open but Not Ambient-Open Is a Valid Boundary Case"],"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-20261108-r03-subspace-topology-0096","batch_id":"batch-20261014-advance-20261108-r03-subspace-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ed0885ae7a020cb349842f64c1b9de6e93b764f6069f6b74fa6b85f677a8858d","family_id":"qf-mod39-subtop-ambient-boundary","seed":202611083096,"semantic_template":"qf-mod39-subtop-ambient-boundary/{\"candidate\":[1,3,4],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,2,4],[0,4],[0]],\"subset\":[1,3,4]}"},"fields":{"title":"Relative-Open but Not Ambient-Open Is a Valid Boundary Case","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 2, 4], [0, 4]], and subset Y=[1, 3, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[1, 3, 4] is a duplicate-free subset of Y. Give (candidate, relative_open, ambient_open, intersection_certificate, boundary).","answer":{"kind":"expression","value":"([1, 3, 4], Yes, No, Yes, O_intersect_Y_not_O_subset_Y)","explanation":"Require the actual candidate to be relatively open but not ambient open. The parent-open intersection certificate does not require its ambient witness U to lie inside Y. The exact result, retaining the requested order, is ([1, 3, 4], Yes, No, Yes, O_intersect_Y_not_O_subset_Y)."},"skills":["Relative-Open but Not Ambient-Open Is a Valid Boundary Case"],"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-20261108-r03-subspace-topology-0097","batch_id":"batch-20261014-advance-20261108-r03-subspace-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2c7041f7f9a67dab96b57215d2cb894156d4b54faec798c98f2bb1d905580ad3","family_id":"qf-mod39-subtop-ambient-boundary","seed":202611083097,"semantic_template":"qf-mod39-subtop-ambient-boundary/{\"candidate\":[1,3,4],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,4],[0,4],[0]],\"subset\":[1,3,4]}"},"fields":{"title":"Relative-Open but Not Ambient-Open Is a Valid Boundary Case","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 2, 3, 4], [0, 2, 4], [0, 4]], and subset Y=[1, 3, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[1, 3, 4] is a duplicate-free subset of Y. Give (candidate, relative_open, ambient_open, intersection_certificate, boundary).","answer":{"kind":"expression","value":"([1, 3, 4], Yes, No, Yes, O_intersect_Y_not_O_subset_Y)","explanation":"Require the actual candidate to be relatively open but not ambient open. The parent-open intersection certificate does not require its ambient witness U to lie inside Y. The exact result, retaining the requested order, is ([1, 3, 4], Yes, No, Yes, O_intersect_Y_not_O_subset_Y)."},"skills":["Relative-Open but Not Ambient-Open Is a Valid Boundary Case"],"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-20261108-r03-subspace-topology-0098","batch_id":"batch-20261014-advance-20261108-r03-subspace-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"48dec3a2fc430b995acddf3fc6fb7e613c0d71c5b0f025fa55f07cf40aa9ddef","family_id":"qf-mod39-subtop-ambient-boundary","seed":202611083098,"semantic_template":"qf-mod39-subtop-ambient-boundary/{\"candidate\":[1,2,4],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,3,4],[0,4],[0]],\"subset\":[1,2,4]}"},"fields":{"title":"Relative-Open but Not Ambient-Open Is a Valid Boundary Case","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 3, 4], [0, 3, 4], [0, 4]], and subset Y=[1, 2, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[1, 2, 4] is a duplicate-free subset of Y. Give (candidate, relative_open, ambient_open, intersection_certificate, boundary).","answer":{"kind":"expression","value":"([1, 2, 4], Yes, No, Yes, O_intersect_Y_not_O_subset_Y)","explanation":"Require the actual candidate to be relatively open but not ambient open. The parent-open intersection certificate does not require its ambient witness U to lie inside Y. The exact result, retaining the requested order, is ([1, 2, 4], Yes, No, Yes, O_intersect_Y_not_O_subset_Y)."},"skills":["Relative-Open but Not Ambient-Open Is a Valid Boundary Case"],"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-20261108-r03-subspace-topology-0099","batch_id":"batch-20261014-advance-20261108-r03-subspace-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eab7839e1e939c9ad008d807a2a40698593f23119da9e88f82f7d9e615546ac1","family_id":"qf-mod39-subtop-ambient-boundary","seed":202611083099,"semantic_template":"qf-mod39-subtop-ambient-boundary/{\"candidate\":[1,2,4],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,3,4],[0,4],[0]],\"subset\":[1,2,4]}"},"fields":{"title":"Relative-Open but Not Ambient-Open Is a Valid Boundary Case","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 2, 3, 4], [0, 3, 4], [0, 4]], and subset Y=[1, 2, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[1, 2, 4] is a duplicate-free subset of Y. Give (candidate, relative_open, ambient_open, intersection_certificate, boundary).","answer":{"kind":"expression","value":"([1, 2, 4], Yes, No, Yes, O_intersect_Y_not_O_subset_Y)","explanation":"Require the actual candidate to be relatively open but not ambient open. The parent-open intersection certificate does not require its ambient witness U to lie inside Y. The exact result, retaining the requested order, is ([1, 2, 4], Yes, No, Yes, O_intersect_Y_not_O_subset_Y)."},"skills":["Relative-Open but Not Ambient-Open Is a Valid Boundary Case"],"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-20261108-r03-subspace-topology-0100","batch_id":"batch-20261014-advance-20261108-r03-subspace-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"74d4d4ea86f785dd9a2dd7fd309768a669d3bfcbbc2219e359a0421fafadd4ec","family_id":"qf-mod39-subtop-ambient-boundary","seed":202611083100,"semantic_template":"qf-mod39-subtop-ambient-boundary/{\"candidate\":[0,3,4],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2],[0,1],[1]],\"subset\":[0,3,4]}"},"fields":{"title":"Relative-Open but Not Ambient-Open Is a Valid Boundary Case","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [1]], and subset Y=[0, 3, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[0, 3, 4] is a duplicate-free subset of Y. Give (candidate, relative_open, ambient_open, intersection_certificate, boundary).","answer":{"kind":"expression","value":"([0, 3, 4], Yes, No, Yes, O_intersect_Y_not_O_subset_Y)","explanation":"Require the actual candidate to be relatively open but not ambient open. The parent-open intersection certificate does not require its ambient witness U to lie inside Y. The exact result, retaining the requested order, is ([0, 3, 4], Yes, No, Yes, O_intersect_Y_not_O_subset_Y)."},"skills":["Relative-Open but Not Ambient-Open Is a Valid Boundary Case"],"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."}}