{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r02-closure-and-interior-0001","batch_id":"batch-20261014-advance-20261101-r02-closure-and-interior-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cf6f5471acd6f5f47ed56eae2b2c1c945a185e529a6d1e4d625b8315995700fd","family_id":"qf-mod32-topology-closure","seed":202611012001,"semantic_template":"qf-mod32-topology-closure/{\"n\":5,\"subset\":[1,2],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2],[0,1],[0]]}"},"fields":{"title":"Closure Intersects All Closed Supersets in the Specified Topology","stem":"Use the complete finite topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[1, 2]. Give (subset,closed_supersets,closure). Closed candidates are complements of the supplied opens in their input order; retain that order among qualifying supersets.","answer":{"kind":"expression","value":"([1, 2], [[0, 1, 2, 3, 4], [1, 2, 3, 4]], [1, 2, 3, 4])","explanation":"Form every closed set relative to this carrier, retain all that contain the subset and intersect them. One selected closed superset is not generally the closure. The exact result, retaining the requested order, is ([1, 2], [[0, 1, 2, 3, 4], [1, 2, 3, 4]], [1, 2, 3, 4])."},"skills":["Closure Intersects All Closed Supersets in the Specified 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-20261101-r02-closure-and-interior-0002","batch_id":"batch-20261014-advance-20261101-r02-closure-and-interior-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f06f77f48b1387abb1942fb5c5c4a057589e265220606cb6513661b698a45675","family_id":"qf-mod32-topology-closure","seed":202611012002,"semantic_template":"qf-mod32-topology-closure/{\"n\":5,\"subset\":[1,2],\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,2],[0,1],[0]]}"},"fields":{"title":"Closure Intersects All Closed Supersets in the Specified Topology","stem":"Use the complete finite topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[1, 2]. Give (subset,closed_supersets,closure). Closed candidates are complements of the supplied opens in their input order; retain that order among qualifying supersets.","answer":{"kind":"expression","value":"([1, 2], [[0, 1, 2, 3, 4], [1, 2, 3, 4]], [1, 2, 3, 4])","explanation":"Form every closed set relative to this carrier, retain all that contain the subset and intersect them. One selected closed superset is not generally the closure. The exact result, retaining the requested order, is ([1, 2], [[0, 1, 2, 3, 4], [1, 2, 3, 4]], [1, 2, 3, 4])."},"skills":["Closure Intersects All Closed Supersets in the Specified 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-20261101-r02-closure-and-interior-0003","batch_id":"batch-20261014-advance-20261101-r02-closure-and-interior-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"424924cc252ed2d9f4460eceafcc93a061d81cf4c637602952a14b1dedff6238","family_id":"qf-mod32-topology-closure","seed":202611012003,"semantic_template":"qf-mod32-topology-closure/{\"n\":5,\"subset\":[1,3],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,3],[0,1],[0]]}"},"fields":{"title":"Closure Intersects All Closed Supersets in the Specified Topology","stem":"Use the complete finite topology=[[], [0], [0, 1], [0, 1, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[1, 3]. Give (subset,closed_supersets,closure). Closed candidates are complements of the supplied opens in their input order; retain that order among qualifying supersets.","answer":{"kind":"expression","value":"([1, 3], [[0, 1, 2, 3, 4], [1, 2, 3, 4]], [1, 2, 3, 4])","explanation":"Form every closed set relative to this carrier, retain all that contain the subset and intersect them. One selected closed superset is not generally the closure. The exact result, retaining the requested order, is ([1, 3], [[0, 1, 2, 3, 4], [1, 2, 3, 4]], [1, 2, 3, 4])."},"skills":["Closure Intersects All Closed Supersets in the Specified 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-20261101-r02-closure-and-interior-0004","batch_id":"batch-20261014-advance-20261101-r02-closure-and-interior-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"490d928f7ea56536a0502a2d7ad1fc7aecb93f896ef716dc2d5bbea7c77093be","family_id":"qf-mod32-topology-closure","seed":202611012004,"semantic_template":"qf-mod32-topology-closure/{\"n\":5,\"subset\":[1,3],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,3],[0,1],[0]]}"},"fields":{"title":"Closure Intersects All Closed Supersets in the Specified Topology","stem":"Use the complete finite topology=[[], [0], [0, 1], [0, 1, 3], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[1, 3]. Give (subset,closed_supersets,closure). Closed candidates are complements of the supplied opens in their input order; retain that order among qualifying supersets.","answer":{"kind":"expression","value":"([1, 3], [[0, 1, 2, 3, 4], [1, 2, 3, 4]], [1, 2, 3, 4])","explanation":"Form every closed set relative to this carrier, retain all that contain the subset and intersect them. One selected closed superset is not generally the closure. The exact result, retaining the requested order, is ([1, 3], [[0, 1, 2, 3, 4], [1, 2, 3, 4]], [1, 2, 3, 4])."},"skills":["Closure Intersects All Closed Supersets in the Specified 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-20261101-r02-closure-and-interior-0005","batch_id":"batch-20261014-advance-20261101-r02-closure-and-interior-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"198d0d83cee79a26e82da7407fd164a9088f4438220a5e3364c712155b9d06a3","family_id":"qf-mod32-topology-closure","seed":202611012005,"semantic_template":"qf-mod32-topology-closure/{\"n\":5,\"subset\":[1,4],\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,4],[0,1],[0]]}"},"fields":{"title":"Closure Intersects All Closed Supersets in the Specified Topology","stem":"Use the complete finite topology=[[], [0], [0, 1], [0, 1, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[1, 4]. Give (subset,closed_supersets,closure). Closed candidates are complements of the supplied opens in their input order; retain that order among qualifying supersets.","answer":{"kind":"expression","value":"([1, 4], [[0, 1, 2, 3, 4], [1, 2, 3, 4]], [1, 2, 3, 4])","explanation":"Form every closed set relative to this carrier, retain all that contain the subset and intersect them. One selected closed superset is not generally the closure. The exact result, retaining the requested order, is ([1, 4], [[0, 1, 2, 3, 4], [1, 2, 3, 4]], [1, 2, 3, 4])."},"skills":["Closure Intersects All Closed Supersets in the Specified 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-20261101-r02-closure-and-interior-0006","batch_id":"batch-20261014-advance-20261101-r02-closure-and-interior-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a372dae7e2fb7b85edc69422a878e15b492a8da185f1c2b49d05c6c29a1de828","family_id":"qf-mod32-topology-closure","seed":202611012006,"semantic_template":"qf-mod32-topology-closure/{\"n\":5,\"subset\":[1,4],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,4],[0,1],[0]]}"},"fields":{"title":"Closure Intersects All Closed Supersets in the Specified Topology","stem":"Use the complete finite topology=[[], [0], [0, 1], [0, 1, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[1, 4]. Give (subset,closed_supersets,closure). Closed candidates are complements of the supplied opens in their input order; retain that order among qualifying supersets.","answer":{"kind":"expression","value":"([1, 4], [[0, 1, 2, 3, 4], [1, 2, 3, 4]], [1, 2, 3, 4])","explanation":"Form every closed set relative to this carrier, retain all that contain the subset and intersect them. One selected closed superset is not generally the closure. The exact result, retaining the requested order, is ([1, 4], [[0, 1, 2, 3, 4], [1, 2, 3, 4]], [1, 2, 3, 4])."},"skills":["Closure Intersects All Closed Supersets in the Specified 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-20261101-r02-closure-and-interior-0007","batch_id":"batch-20261014-advance-20261101-r02-closure-and-interior-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5e75f7d4a642ee7619a2895b90240ed7b3d72d60ec2a115af45bd328d714c172","family_id":"qf-mod32-topology-closure","seed":202611012007,"semantic_template":"qf-mod32-topology-closure/{\"n\":5,\"subset\":[1,2],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2],[0,2],[0]]}"},"fields":{"title":"Closure Intersects All Closed Supersets in the Specified Topology","stem":"Use the complete finite topology=[[], [0], [0, 2], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[1, 2]. Give (subset,closed_supersets,closure). Closed candidates are complements of the supplied opens in their input order; retain that order among qualifying supersets.","answer":{"kind":"expression","value":"([1, 2], [[0, 1, 2, 3, 4], [1, 2, 3, 4]], [1, 2, 3, 4])","explanation":"Form every closed set relative to this carrier, retain all that contain the subset and intersect them. One selected closed superset is not generally the closure. The exact result, retaining the requested order, is ([1, 2], [[0, 1, 2, 3, 4], [1, 2, 3, 4]], [1, 2, 3, 4])."},"skills":["Closure Intersects All Closed Supersets in the Specified 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-20261101-r02-closure-and-interior-0008","batch_id":"batch-20261014-advance-20261101-r02-closure-and-interior-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4adaa412cbcf7c0e44a10dcf825901de3dd8e557ef4e23d8b8b6224b13e4e0a2","family_id":"qf-mod32-topology-closure","seed":202611012008,"semantic_template":"qf-mod32-topology-closure/{\"n\":5,\"subset\":[1,2],\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,2],[0,2],[0]]}"},"fields":{"title":"Closure Intersects All Closed Supersets in the Specified Topology","stem":"Use the complete finite topology=[[], [0], [0, 2], [0, 1, 2], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[1, 2]. Give (subset,closed_supersets,closure). Closed candidates are complements of the supplied opens in their input order; retain that order among qualifying supersets.","answer":{"kind":"expression","value":"([1, 2], [[0, 1, 2, 3, 4], [1, 2, 3, 4]], [1, 2, 3, 4])","explanation":"Form every closed set relative to this carrier, retain all that contain the subset and intersect them. One selected closed superset is not generally the closure. The exact result, retaining the requested order, is ([1, 2], [[0, 1, 2, 3, 4], [1, 2, 3, 4]], [1, 2, 3, 4])."},"skills":["Closure Intersects All Closed Supersets in the Specified 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-20261101-r02-closure-and-interior-0009","batch_id":"batch-20261014-advance-20261101-r02-closure-and-interior-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fab2bf20509d5c98029e0e799ab5f1f282ca9d20da124ab1900e0e3d37e323de","family_id":"qf-mod32-topology-closure","seed":202611012009,"semantic_template":"qf-mod32-topology-closure/{\"n\":5,\"subset\":[2,3],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,2,3],[0,2],[0]]}"},"fields":{"title":"Closure Intersects All Closed Supersets in the Specified Topology","stem":"Use the complete finite topology=[[], [0], [0, 2], [0, 2, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[2, 3]. Give (subset,closed_supersets,closure). Closed candidates are complements of the supplied opens in their input order; retain that order among qualifying supersets.","answer":{"kind":"expression","value":"([2, 3], [[0, 1, 2, 3, 4], [1, 2, 3, 4]], [1, 2, 3, 4])","explanation":"Form every closed set relative to this carrier, retain all that contain the subset and intersect them. One selected closed superset is not generally the closure. The exact result, retaining the requested order, is ([2, 3], [[0, 1, 2, 3, 4], [1, 2, 3, 4]], [1, 2, 3, 4])."},"skills":["Closure Intersects All Closed Supersets in the Specified 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-20261101-r02-closure-and-interior-0010","batch_id":"batch-20261014-advance-20261101-r02-closure-and-interior-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"27effd9e9fcd6c54de4585fddcd175d3b279c7003bd62056aa4978e0910277b3","family_id":"qf-mod32-topology-closure","seed":202611012010,"semantic_template":"qf-mod32-topology-closure/{\"n\":5,\"subset\":[2,3],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,3],[0,2],[0]]}"},"fields":{"title":"Closure Intersects All Closed Supersets in the Specified Topology","stem":"Use the complete finite topology=[[], [0], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[2, 3]. Give (subset,closed_supersets,closure). Closed candidates are complements of the supplied opens in their input order; retain that order among qualifying supersets.","answer":{"kind":"expression","value":"([2, 3], [[0, 1, 2, 3, 4], [1, 2, 3, 4]], [1, 2, 3, 4])","explanation":"Form every closed set relative to this carrier, retain all that contain the subset and intersect them. One selected closed superset is not generally the closure. The exact result, retaining the requested order, is ([2, 3], [[0, 1, 2, 3, 4], [1, 2, 3, 4]], [1, 2, 3, 4])."},"skills":["Closure Intersects All Closed Supersets in the Specified 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-20261101-r02-closure-and-interior-0011","batch_id":"batch-20261014-advance-20261101-r02-closure-and-interior-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1b63f095f4a72510c6709e029bb8c7871d1e8e3b6deb8e2a055961c4e1fa230b","family_id":"qf-mod32-topology-closure","seed":202611012011,"semantic_template":"qf-mod32-topology-closure/{\"n\":5,\"subset\":[2,4],\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,2,4],[0,2],[0]]}"},"fields":{"title":"Closure Intersects All Closed Supersets in the Specified Topology","stem":"Use the complete finite topology=[[], [0], [0, 2], [0, 2, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[2, 4]. Give (subset,closed_supersets,closure). Closed candidates are complements of the supplied opens in their input order; retain that order among qualifying supersets.","answer":{"kind":"expression","value":"([2, 4], [[0, 1, 2, 3, 4], [1, 2, 3, 4]], [1, 2, 3, 4])","explanation":"Form every closed set relative to this carrier, retain all that contain the subset and intersect them. One selected closed superset is not generally the closure. The exact result, retaining the requested order, is ([2, 4], [[0, 1, 2, 3, 4], [1, 2, 3, 4]], [1, 2, 3, 4])."},"skills":["Closure Intersects All Closed Supersets in the Specified 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-20261101-r02-closure-and-interior-0012","batch_id":"batch-20261014-advance-20261101-r02-closure-and-interior-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ee719b22849270062d04072df13c382345fa749278bedb61fd32b687414a24cd","family_id":"qf-mod32-topology-closure","seed":202611012012,"semantic_template":"qf-mod32-topology-closure/{\"n\":5,\"subset\":[2,4],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,4],[0,2],[0]]}"},"fields":{"title":"Closure Intersects All Closed Supersets in the Specified Topology","stem":"Use the complete finite topology=[[], [0], [0, 2], [0, 2, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[2, 4]. Give (subset,closed_supersets,closure). Closed candidates are complements of the supplied opens in their input order; retain that order among qualifying supersets.","answer":{"kind":"expression","value":"([2, 4], [[0, 1, 2, 3, 4], [1, 2, 3, 4]], [1, 2, 3, 4])","explanation":"Form every closed set relative to this carrier, retain all that contain the subset and intersect them. One selected closed superset is not generally the closure. The exact result, retaining the requested order, is ([2, 4], [[0, 1, 2, 3, 4], [1, 2, 3, 4]], [1, 2, 3, 4])."},"skills":["Closure Intersects All Closed Supersets in the Specified 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-20261101-r02-closure-and-interior-0013","batch_id":"batch-20261014-advance-20261101-r02-closure-and-interior-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2dc51f1473046702ae5e553c75479e98b3e5dc5dfec3c5b294ac21f6d6cbde3f","family_id":"qf-mod32-topology-closure","seed":202611012013,"semantic_template":"qf-mod32-topology-closure/{\"n\":5,\"subset\":[1,3],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,3],[0,3],[0]]}"},"fields":{"title":"Closure Intersects All Closed Supersets in the Specified Topology","stem":"Use the complete finite topology=[[], [0], [0, 3], [0, 1, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[1, 3]. Give (subset,closed_supersets,closure). Closed candidates are complements of the supplied opens in their input order; retain that order among qualifying supersets.","answer":{"kind":"expression","value":"([1, 3], [[0, 1, 2, 3, 4], [1, 2, 3, 4]], [1, 2, 3, 4])","explanation":"Form every closed set relative to this carrier, retain all that contain the subset and intersect them. One selected closed superset is not generally the closure. The exact result, retaining the requested order, is ([1, 3], [[0, 1, 2, 3, 4], [1, 2, 3, 4]], [1, 2, 3, 4])."},"skills":["Closure Intersects All Closed Supersets in the Specified 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-20261101-r02-closure-and-interior-0014","batch_id":"batch-20261014-advance-20261101-r02-closure-and-interior-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0d8607b90e6750de503dac0f669b2f80647a5de7e9dbe1d252b8c9ce8cb0ad0c","family_id":"qf-mod32-topology-closure","seed":202611012014,"semantic_template":"qf-mod32-topology-closure/{\"n\":5,\"subset\":[1,3],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,3],[0,3],[0]]}"},"fields":{"title":"Closure Intersects All Closed Supersets in the Specified Topology","stem":"Use the complete finite topology=[[], [0], [0, 3], [0, 1, 3], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[1, 3]. Give (subset,closed_supersets,closure). Closed candidates are complements of the supplied opens in their input order; retain that order among qualifying supersets.","answer":{"kind":"expression","value":"([1, 3], [[0, 1, 2, 3, 4], [1, 2, 3, 4]], [1, 2, 3, 4])","explanation":"Form every closed set relative to this carrier, retain all that contain the subset and intersect them. One selected closed superset is not generally the closure. The exact result, retaining the requested order, is ([1, 3], [[0, 1, 2, 3, 4], [1, 2, 3, 4]], [1, 2, 3, 4])."},"skills":["Closure Intersects All Closed Supersets in the Specified 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-20261101-r02-closure-and-interior-0015","batch_id":"batch-20261014-advance-20261101-r02-closure-and-interior-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"701aa0b57f3a00023717d52e8f275d6a7adfafb171843cc7494acd0a78ac0d9c","family_id":"qf-mod32-topology-closure","seed":202611012015,"semantic_template":"qf-mod32-topology-closure/{\"n\":5,\"subset\":[2,3],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,2,3],[0,3],[0]]}"},"fields":{"title":"Closure Intersects All Closed Supersets in the Specified Topology","stem":"Use the complete finite topology=[[], [0], [0, 3], [0, 2, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[2, 3]. Give (subset,closed_supersets,closure). Closed candidates are complements of the supplied opens in their input order; retain that order among qualifying supersets.","answer":{"kind":"expression","value":"([2, 3], [[0, 1, 2, 3, 4], [1, 2, 3, 4]], [1, 2, 3, 4])","explanation":"Form every closed set relative to this carrier, retain all that contain the subset and intersect them. One selected closed superset is not generally the closure. The exact result, retaining the requested order, is ([2, 3], [[0, 1, 2, 3, 4], [1, 2, 3, 4]], [1, 2, 3, 4])."},"skills":["Closure Intersects All Closed Supersets in the Specified 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-20261101-r02-closure-and-interior-0016","batch_id":"batch-20261014-advance-20261101-r02-closure-and-interior-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1d151d20d2b49e7229e72b20ac8b7c097fd0ccffcd409a7c46da42f044ab2a29","family_id":"qf-mod32-topology-closure","seed":202611012016,"semantic_template":"qf-mod32-topology-closure/{\"n\":5,\"subset\":[2,3],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,3],[0,3],[0]]}"},"fields":{"title":"Closure Intersects All Closed Supersets in the Specified Topology","stem":"Use the complete finite topology=[[], [0], [0, 3], [0, 2, 3], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[2, 3]. Give (subset,closed_supersets,closure). Closed candidates are complements of the supplied opens in their input order; retain that order among qualifying supersets.","answer":{"kind":"expression","value":"([2, 3], [[0, 1, 2, 3, 4], [1, 2, 3, 4]], [1, 2, 3, 4])","explanation":"Form every closed set relative to this carrier, retain all that contain the subset and intersect them. One selected closed superset is not generally the closure. The exact result, retaining the requested order, is ([2, 3], [[0, 1, 2, 3, 4], [1, 2, 3, 4]], [1, 2, 3, 4])."},"skills":["Closure Intersects All Closed Supersets in the Specified 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-20261101-r02-closure-and-interior-0017","batch_id":"batch-20261014-advance-20261101-r02-closure-and-interior-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0bcfd409715beea7685e8cd776857b21e1691a964b246d734512f70dd1532ecf","family_id":"qf-mod32-topology-closure","seed":202611012017,"semantic_template":"qf-mod32-topology-closure/{\"n\":5,\"subset\":[3,4],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,3,4],[0,3],[0]]}"},"fields":{"title":"Closure Intersects All Closed Supersets in the Specified Topology","stem":"Use the complete finite topology=[[], [0], [0, 3], [0, 3, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[3, 4]. Give (subset,closed_supersets,closure). Closed candidates are complements of the supplied opens in their input order; retain that order among qualifying supersets.","answer":{"kind":"expression","value":"([3, 4], [[0, 1, 2, 3, 4], [1, 2, 3, 4]], [1, 2, 3, 4])","explanation":"Form every closed set relative to this carrier, retain all that contain the subset and intersect them. One selected closed superset is not generally the closure. The exact result, retaining the requested order, is ([3, 4], [[0, 1, 2, 3, 4], [1, 2, 3, 4]], [1, 2, 3, 4])."},"skills":["Closure Intersects All Closed Supersets in the Specified 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-20261101-r02-closure-and-interior-0018","batch_id":"batch-20261014-advance-20261101-r02-closure-and-interior-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"14c9e086b23aec9387f028e9edbb83f114dc9f4bae8e3a57bb9f15506c04008c","family_id":"qf-mod32-topology-closure","seed":202611012018,"semantic_template":"qf-mod32-topology-closure/{\"n\":5,\"subset\":[3,4],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,3,4],[0,3],[0]]}"},"fields":{"title":"Closure Intersects All Closed Supersets in the Specified Topology","stem":"Use the complete finite topology=[[], [0], [0, 3], [0, 3, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[3, 4]. Give (subset,closed_supersets,closure). Closed candidates are complements of the supplied opens in their input order; retain that order among qualifying supersets.","answer":{"kind":"expression","value":"([3, 4], [[0, 1, 2, 3, 4], [1, 2, 3, 4]], [1, 2, 3, 4])","explanation":"Form every closed set relative to this carrier, retain all that contain the subset and intersect them. One selected closed superset is not generally the closure. The exact result, retaining the requested order, is ([3, 4], [[0, 1, 2, 3, 4], [1, 2, 3, 4]], [1, 2, 3, 4])."},"skills":["Closure Intersects All Closed Supersets in the Specified 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-20261101-r02-closure-and-interior-0019","batch_id":"batch-20261014-advance-20261101-r02-closure-and-interior-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6ecddf40207e3008fe446c665428611a53295d1846cb1cd7942373cff54e8a5a","family_id":"qf-mod32-topology-closure","seed":202611012019,"semantic_template":"qf-mod32-topology-closure/{\"n\":5,\"subset\":[1,4],\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,4],[0,4],[0]]}"},"fields":{"title":"Closure Intersects All Closed Supersets in the Specified Topology","stem":"Use the complete finite topology=[[], [0], [0, 4], [0, 1, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[1, 4]. Give (subset,closed_supersets,closure). Closed candidates are complements of the supplied opens in their input order; retain that order among qualifying supersets.","answer":{"kind":"expression","value":"([1, 4], [[0, 1, 2, 3, 4], [1, 2, 3, 4]], [1, 2, 3, 4])","explanation":"Form every closed set relative to this carrier, retain all that contain the subset and intersect them. One selected closed superset is not generally the closure. The exact result, retaining the requested order, is ([1, 4], [[0, 1, 2, 3, 4], [1, 2, 3, 4]], [1, 2, 3, 4])."},"skills":["Closure Intersects All Closed Supersets in the Specified 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-20261101-r02-closure-and-interior-0020","batch_id":"batch-20261014-advance-20261101-r02-closure-and-interior-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7557e5f46a0eba82ad1b9b5ead5bbfabe6e7f5c7cb3f7d1f3c87d5d1a20c5515","family_id":"qf-mod32-topology-closure","seed":202611012020,"semantic_template":"qf-mod32-topology-closure/{\"n\":5,\"subset\":[1,4],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,4],[0,4],[0]]}"},"fields":{"title":"Closure Intersects All Closed Supersets in the Specified Topology","stem":"Use the complete finite topology=[[], [0], [0, 4], [0, 1, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[1, 4]. Give (subset,closed_supersets,closure). Closed candidates are complements of the supplied opens in their input order; retain that order among qualifying supersets.","answer":{"kind":"expression","value":"([1, 4], [[0, 1, 2, 3, 4], [1, 2, 3, 4]], [1, 2, 3, 4])","explanation":"Form every closed set relative to this carrier, retain all that contain the subset and intersect them. One selected closed superset is not generally the closure. The exact result, retaining the requested order, is ([1, 4], [[0, 1, 2, 3, 4], [1, 2, 3, 4]], [1, 2, 3, 4])."},"skills":["Closure Intersects All Closed Supersets in the Specified 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-20261101-r02-closure-and-interior-0021","batch_id":"batch-20261014-advance-20261101-r02-closure-and-interior-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"256b1fd7f192dde4e2e46377b30ea4f4c8d73f64a444db8606118458b16c5789","family_id":"qf-mod32-topology-closure","seed":202611012021,"semantic_template":"qf-mod32-topology-closure/{\"n\":5,\"subset\":[2,4],\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,2,4],[0,4],[0]]}"},"fields":{"title":"Closure Intersects All Closed Supersets in the Specified Topology","stem":"Use the complete finite topology=[[], [0], [0, 4], [0, 2, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[2, 4]. Give (subset,closed_supersets,closure). Closed candidates are complements of the supplied opens in their input order; retain that order among qualifying supersets.","answer":{"kind":"expression","value":"([2, 4], [[0, 1, 2, 3, 4], [1, 2, 3, 4]], [1, 2, 3, 4])","explanation":"Form every closed set relative to this carrier, retain all that contain the subset and intersect them. One selected closed superset is not generally the closure. The exact result, retaining the requested order, is ([2, 4], [[0, 1, 2, 3, 4], [1, 2, 3, 4]], [1, 2, 3, 4])."},"skills":["Closure Intersects All Closed Supersets in the Specified 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-20261101-r02-closure-and-interior-0022","batch_id":"batch-20261014-advance-20261101-r02-closure-and-interior-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5a72349f55c292f2e15cd89520971968fa3cad4feba3c2b0b10a57c01adbe755","family_id":"qf-mod32-topology-closure","seed":202611012022,"semantic_template":"qf-mod32-topology-closure/{\"n\":5,\"subset\":[2,4],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,4],[0,4],[0]]}"},"fields":{"title":"Closure Intersects All Closed Supersets in the Specified Topology","stem":"Use the complete finite topology=[[], [0], [0, 4], [0, 2, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[2, 4]. Give (subset,closed_supersets,closure). Closed candidates are complements of the supplied opens in their input order; retain that order among qualifying supersets.","answer":{"kind":"expression","value":"([2, 4], [[0, 1, 2, 3, 4], [1, 2, 3, 4]], [1, 2, 3, 4])","explanation":"Form every closed set relative to this carrier, retain all that contain the subset and intersect them. One selected closed superset is not generally the closure. The exact result, retaining the requested order, is ([2, 4], [[0, 1, 2, 3, 4], [1, 2, 3, 4]], [1, 2, 3, 4])."},"skills":["Closure Intersects All Closed Supersets in the Specified 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-20261101-r02-closure-and-interior-0023","batch_id":"batch-20261014-advance-20261101-r02-closure-and-interior-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e5d9c11d78d5b82cc60b050d4b3e6af2ba27393f2c12edb8bae9e6634694c522","family_id":"qf-mod32-topology-closure","seed":202611012023,"semantic_template":"qf-mod32-topology-closure/{\"n\":5,\"subset\":[3,4],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,3,4],[0,4],[0]]}"},"fields":{"title":"Closure Intersects All Closed Supersets in the Specified Topology","stem":"Use the complete finite topology=[[], [0], [0, 4], [0, 3, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[3, 4]. Give (subset,closed_supersets,closure). Closed candidates are complements of the supplied opens in their input order; retain that order among qualifying supersets.","answer":{"kind":"expression","value":"([3, 4], [[0, 1, 2, 3, 4], [1, 2, 3, 4]], [1, 2, 3, 4])","explanation":"Form every closed set relative to this carrier, retain all that contain the subset and intersect them. One selected closed superset is not generally the closure. The exact result, retaining the requested order, is ([3, 4], [[0, 1, 2, 3, 4], [1, 2, 3, 4]], [1, 2, 3, 4])."},"skills":["Closure Intersects All Closed Supersets in the Specified 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-20261101-r02-closure-and-interior-0024","batch_id":"batch-20261014-advance-20261101-r02-closure-and-interior-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c272447a194b69833d80bcf5335fa2073e615c027161bdd4d3d86b6b6a6b4b1a","family_id":"qf-mod32-topology-closure","seed":202611012024,"semantic_template":"qf-mod32-topology-closure/{\"n\":5,\"subset\":[3,4],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,3,4],[0,4],[0]]}"},"fields":{"title":"Closure Intersects All Closed Supersets in the Specified Topology","stem":"Use the complete finite topology=[[], [0], [0, 4], [0, 3, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[3, 4]. Give (subset,closed_supersets,closure). Closed candidates are complements of the supplied opens in their input order; retain that order among qualifying supersets.","answer":{"kind":"expression","value":"([3, 4], [[0, 1, 2, 3, 4], [1, 2, 3, 4]], [1, 2, 3, 4])","explanation":"Form every closed set relative to this carrier, retain all that contain the subset and intersect them. One selected closed superset is not generally the closure. The exact result, retaining the requested order, is ([3, 4], [[0, 1, 2, 3, 4], [1, 2, 3, 4]], [1, 2, 3, 4])."},"skills":["Closure Intersects All Closed Supersets in the Specified 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-20261101-r02-closure-and-interior-0025","batch_id":"batch-20261014-advance-20261101-r02-closure-and-interior-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4e93803d24402f4e3003a95a4d587f91900494dd43925e75e953d6b91c1e2e44","family_id":"qf-mod32-topology-closure","seed":202611012025,"semantic_template":"qf-mod32-topology-closure/{\"n\":5,\"subset\":[0,2],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2],[0,1],[1]]}"},"fields":{"title":"Closure Intersects All Closed Supersets in the Specified Topology","stem":"Use the complete finite topology=[[], [1], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[0, 2]. Give (subset,closed_supersets,closure). Closed candidates are complements of the supplied opens in their input order; retain that order among qualifying supersets.","answer":{"kind":"expression","value":"([0, 2], [[0, 1, 2, 3, 4], [0, 2, 3, 4]], [0, 2, 3, 4])","explanation":"Form every closed set relative to this carrier, retain all that contain the subset and intersect them. One selected closed superset is not generally the closure. The exact result, retaining the requested order, is ([0, 2], [[0, 1, 2, 3, 4], [0, 2, 3, 4]], [0, 2, 3, 4])."},"skills":["Closure Intersects All Closed Supersets in the Specified 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-20261101-r02-closure-and-interior-0026","batch_id":"batch-20261014-advance-20261101-r02-closure-and-interior-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4a70072e25c5cdf43762521602c3721072ba02837a9e4f3e46cca20cff26ce90","family_id":"qf-mod32-topology-interior","seed":202611012026,"semantic_template":"qf-mod32-topology-interior/{\"n\":5,\"subset\":[1,2],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2],[0,1],[0]]}"},"fields":{"title":"Interior Unites All Open Subsets, Not One Convenient Open Set","stem":"Use the complete finite topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[1, 2]. Give (subset,open_subsets,interior), retaining original topology order among the qualifying opens.","answer":{"kind":"expression","value":"([1, 2], [[]], [])","explanation":"Select every open contained in the subset and take their union in increasing carrier order. The result is relative to this exact topology, not an unstated metric. The exact result, retaining the requested order, is ([1, 2], [[]], [])."},"skills":["Interior Unites All Open Subsets, Not One Convenient Open Set"],"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-20261101-r02-closure-and-interior-0027","batch_id":"batch-20261014-advance-20261101-r02-closure-and-interior-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"072f92534dac5d3369555c93ffdfda050f56d25be6472eb6117816f9ec078032","family_id":"qf-mod32-topology-interior","seed":202611012027,"semantic_template":"qf-mod32-topology-interior/{\"n\":5,\"subset\":[1,2],\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,2],[0,1],[0]]}"},"fields":{"title":"Interior Unites All Open Subsets, Not One Convenient Open Set","stem":"Use the complete finite topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[1, 2]. Give (subset,open_subsets,interior), retaining original topology order among the qualifying opens.","answer":{"kind":"expression","value":"([1, 2], [[]], [])","explanation":"Select every open contained in the subset and take their union in increasing carrier order. The result is relative to this exact topology, not an unstated metric. The exact result, retaining the requested order, is ([1, 2], [[]], [])."},"skills":["Interior Unites All Open Subsets, Not One Convenient Open Set"],"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-20261101-r02-closure-and-interior-0028","batch_id":"batch-20261014-advance-20261101-r02-closure-and-interior-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7731a47db977349f0af6aa0810810a8c260eb2eb45ce4ca52d6ac8951cb2fa10","family_id":"qf-mod32-topology-interior","seed":202611012028,"semantic_template":"qf-mod32-topology-interior/{\"n\":5,\"subset\":[1,3],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,3],[0,1],[0]]}"},"fields":{"title":"Interior Unites All Open Subsets, Not One Convenient Open Set","stem":"Use the complete finite topology=[[], [0], [0, 1], [0, 1, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[1, 3]. Give (subset,open_subsets,interior), retaining original topology order among the qualifying opens.","answer":{"kind":"expression","value":"([1, 3], [[]], [])","explanation":"Select every open contained in the subset and take their union in increasing carrier order. The result is relative to this exact topology, not an unstated metric. The exact result, retaining the requested order, is ([1, 3], [[]], [])."},"skills":["Interior Unites All Open Subsets, Not One Convenient Open Set"],"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-20261101-r02-closure-and-interior-0029","batch_id":"batch-20261014-advance-20261101-r02-closure-and-interior-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b1a7bc9a9d6d2ffd612872d1f2e5de5433e8041c70e54782a4d46a4dbd496df7","family_id":"qf-mod32-topology-interior","seed":202611012029,"semantic_template":"qf-mod32-topology-interior/{\"n\":5,\"subset\":[1,3],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,3],[0,1],[0]]}"},"fields":{"title":"Interior Unites All Open Subsets, Not One Convenient Open Set","stem":"Use the complete finite topology=[[], [0], [0, 1], [0, 1, 3], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[1, 3]. Give (subset,open_subsets,interior), retaining original topology order among the qualifying opens.","answer":{"kind":"expression","value":"([1, 3], [[]], [])","explanation":"Select every open contained in the subset and take their union in increasing carrier order. The result is relative to this exact topology, not an unstated metric. The exact result, retaining the requested order, is ([1, 3], [[]], [])."},"skills":["Interior Unites All Open Subsets, Not One Convenient Open Set"],"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-20261101-r02-closure-and-interior-0030","batch_id":"batch-20261014-advance-20261101-r02-closure-and-interior-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6f579a479aea6789b470250034706b8c1abbe90af6c8ab447e597b31c56ba8d4","family_id":"qf-mod32-topology-interior","seed":202611012030,"semantic_template":"qf-mod32-topology-interior/{\"n\":5,\"subset\":[1,4],\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,4],[0,1],[0]]}"},"fields":{"title":"Interior Unites All Open Subsets, Not One Convenient Open Set","stem":"Use the complete finite topology=[[], [0], [0, 1], [0, 1, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[1, 4]. Give (subset,open_subsets,interior), retaining original topology order among the qualifying opens.","answer":{"kind":"expression","value":"([1, 4], [[]], [])","explanation":"Select every open contained in the subset and take their union in increasing carrier order. The result is relative to this exact topology, not an unstated metric. The exact result, retaining the requested order, is ([1, 4], [[]], [])."},"skills":["Interior Unites All Open Subsets, Not One Convenient Open Set"],"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-20261101-r02-closure-and-interior-0031","batch_id":"batch-20261014-advance-20261101-r02-closure-and-interior-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"57f887c311523e8a277094ee0caef8d98ee5918060a6593274479a3c31888974","family_id":"qf-mod32-topology-interior","seed":202611012031,"semantic_template":"qf-mod32-topology-interior/{\"n\":5,\"subset\":[1,4],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,4],[0,1],[0]]}"},"fields":{"title":"Interior Unites All Open Subsets, Not One Convenient Open Set","stem":"Use the complete finite topology=[[], [0], [0, 1], [0, 1, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[1, 4]. Give (subset,open_subsets,interior), retaining original topology order among the qualifying opens.","answer":{"kind":"expression","value":"([1, 4], [[]], [])","explanation":"Select every open contained in the subset and take their union in increasing carrier order. The result is relative to this exact topology, not an unstated metric. The exact result, retaining the requested order, is ([1, 4], [[]], [])."},"skills":["Interior Unites All Open Subsets, Not One Convenient Open Set"],"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-20261101-r02-closure-and-interior-0032","batch_id":"batch-20261014-advance-20261101-r02-closure-and-interior-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ea9cc4f8cef7afedfde5d8f0b8751a053be8e75c85388fa9d68962e4d40e20d6","family_id":"qf-mod32-topology-interior","seed":202611012032,"semantic_template":"qf-mod32-topology-interior/{\"n\":5,\"subset\":[1,2],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2],[0,2],[0]]}"},"fields":{"title":"Interior Unites All Open Subsets, Not One Convenient Open Set","stem":"Use the complete finite topology=[[], [0], [0, 2], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[1, 2]. Give (subset,open_subsets,interior), retaining original topology order among the qualifying opens.","answer":{"kind":"expression","value":"([1, 2], [[]], [])","explanation":"Select every open contained in the subset and take their union in increasing carrier order. The result is relative to this exact topology, not an unstated metric. The exact result, retaining the requested order, is ([1, 2], [[]], [])."},"skills":["Interior Unites All Open Subsets, Not One Convenient Open Set"],"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-20261101-r02-closure-and-interior-0033","batch_id":"batch-20261014-advance-20261101-r02-closure-and-interior-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"662bcae11ddd31865d7afd556f8386789091e745812fccb78c9d812590b54647","family_id":"qf-mod32-topology-interior","seed":202611012033,"semantic_template":"qf-mod32-topology-interior/{\"n\":5,\"subset\":[1,2],\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,2],[0,2],[0]]}"},"fields":{"title":"Interior Unites All Open Subsets, Not One Convenient Open Set","stem":"Use the complete finite topology=[[], [0], [0, 2], [0, 1, 2], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[1, 2]. Give (subset,open_subsets,interior), retaining original topology order among the qualifying opens.","answer":{"kind":"expression","value":"([1, 2], [[]], [])","explanation":"Select every open contained in the subset and take their union in increasing carrier order. The result is relative to this exact topology, not an unstated metric. The exact result, retaining the requested order, is ([1, 2], [[]], [])."},"skills":["Interior Unites All Open Subsets, Not One Convenient Open Set"],"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-20261101-r02-closure-and-interior-0034","batch_id":"batch-20261014-advance-20261101-r02-closure-and-interior-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c2525a5dac167f3785d9f2d69cc85c32741d019c5373d3517e7e65e7be08a6be","family_id":"qf-mod32-topology-interior","seed":202611012034,"semantic_template":"qf-mod32-topology-interior/{\"n\":5,\"subset\":[2,3],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,2,3],[0,2],[0]]}"},"fields":{"title":"Interior Unites All Open Subsets, Not One Convenient Open Set","stem":"Use the complete finite topology=[[], [0], [0, 2], [0, 2, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[2, 3]. Give (subset,open_subsets,interior), retaining original topology order among the qualifying opens.","answer":{"kind":"expression","value":"([2, 3], [[]], [])","explanation":"Select every open contained in the subset and take their union in increasing carrier order. The result is relative to this exact topology, not an unstated metric. The exact result, retaining the requested order, is ([2, 3], [[]], [])."},"skills":["Interior Unites All Open Subsets, Not One Convenient Open Set"],"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-20261101-r02-closure-and-interior-0035","batch_id":"batch-20261014-advance-20261101-r02-closure-and-interior-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6bbda0fdf5fbb7813644b1c8d9e81f0c929e613813690ab794de7693b6d2e6bd","family_id":"qf-mod32-topology-interior","seed":202611012035,"semantic_template":"qf-mod32-topology-interior/{\"n\":5,\"subset\":[2,3],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,3],[0,2],[0]]}"},"fields":{"title":"Interior Unites All Open Subsets, Not One Convenient Open Set","stem":"Use the complete finite topology=[[], [0], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[2, 3]. Give (subset,open_subsets,interior), retaining original topology order among the qualifying opens.","answer":{"kind":"expression","value":"([2, 3], [[]], [])","explanation":"Select every open contained in the subset and take their union in increasing carrier order. The result is relative to this exact topology, not an unstated metric. The exact result, retaining the requested order, is ([2, 3], [[]], [])."},"skills":["Interior Unites All Open Subsets, Not One Convenient Open Set"],"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-20261101-r02-closure-and-interior-0036","batch_id":"batch-20261014-advance-20261101-r02-closure-and-interior-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bac2decf520e9dc05cda21a96ab12e4735e592fce40b823918518b32b111400e","family_id":"qf-mod32-topology-interior","seed":202611012036,"semantic_template":"qf-mod32-topology-interior/{\"n\":5,\"subset\":[2,4],\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,2,4],[0,2],[0]]}"},"fields":{"title":"Interior Unites All Open Subsets, Not One Convenient Open Set","stem":"Use the complete finite topology=[[], [0], [0, 2], [0, 2, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[2, 4]. Give (subset,open_subsets,interior), retaining original topology order among the qualifying opens.","answer":{"kind":"expression","value":"([2, 4], [[]], [])","explanation":"Select every open contained in the subset and take their union in increasing carrier order. The result is relative to this exact topology, not an unstated metric. The exact result, retaining the requested order, is ([2, 4], [[]], [])."},"skills":["Interior Unites All Open Subsets, Not One Convenient Open Set"],"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-20261101-r02-closure-and-interior-0037","batch_id":"batch-20261014-advance-20261101-r02-closure-and-interior-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"30114d59baba14077e2fe4afee19e68776b507fbd438edc1014ca7872d250da0","family_id":"qf-mod32-topology-interior","seed":202611012037,"semantic_template":"qf-mod32-topology-interior/{\"n\":5,\"subset\":[2,4],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,4],[0,2],[0]]}"},"fields":{"title":"Interior Unites All Open Subsets, Not One Convenient Open Set","stem":"Use the complete finite topology=[[], [0], [0, 2], [0, 2, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[2, 4]. Give (subset,open_subsets,interior), retaining original topology order among the qualifying opens.","answer":{"kind":"expression","value":"([2, 4], [[]], [])","explanation":"Select every open contained in the subset and take their union in increasing carrier order. The result is relative to this exact topology, not an unstated metric. The exact result, retaining the requested order, is ([2, 4], [[]], [])."},"skills":["Interior Unites All Open Subsets, Not One Convenient Open Set"],"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-20261101-r02-closure-and-interior-0038","batch_id":"batch-20261014-advance-20261101-r02-closure-and-interior-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e5e226e7e0eaffe447446a201e6af5a560b98e8749909723fd57619526030d34","family_id":"qf-mod32-topology-interior","seed":202611012038,"semantic_template":"qf-mod32-topology-interior/{\"n\":5,\"subset\":[1,3],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,3],[0,3],[0]]}"},"fields":{"title":"Interior Unites All Open Subsets, Not One Convenient Open Set","stem":"Use the complete finite topology=[[], [0], [0, 3], [0, 1, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[1, 3]. Give (subset,open_subsets,interior), retaining original topology order among the qualifying opens.","answer":{"kind":"expression","value":"([1, 3], [[]], [])","explanation":"Select every open contained in the subset and take their union in increasing carrier order. The result is relative to this exact topology, not an unstated metric. The exact result, retaining the requested order, is ([1, 3], [[]], [])."},"skills":["Interior Unites All Open Subsets, Not One Convenient Open Set"],"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-20261101-r02-closure-and-interior-0039","batch_id":"batch-20261014-advance-20261101-r02-closure-and-interior-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0fa6cb7e4fb7c486051f0743b40ade1c5110ec7659b71df45f69563f5bcd60bb","family_id":"qf-mod32-topology-interior","seed":202611012039,"semantic_template":"qf-mod32-topology-interior/{\"n\":5,\"subset\":[1,3],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,3],[0,3],[0]]}"},"fields":{"title":"Interior Unites All Open Subsets, Not One Convenient Open Set","stem":"Use the complete finite topology=[[], [0], [0, 3], [0, 1, 3], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[1, 3]. Give (subset,open_subsets,interior), retaining original topology order among the qualifying opens.","answer":{"kind":"expression","value":"([1, 3], [[]], [])","explanation":"Select every open contained in the subset and take their union in increasing carrier order. The result is relative to this exact topology, not an unstated metric. The exact result, retaining the requested order, is ([1, 3], [[]], [])."},"skills":["Interior Unites All Open Subsets, Not One Convenient Open Set"],"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-20261101-r02-closure-and-interior-0040","batch_id":"batch-20261014-advance-20261101-r02-closure-and-interior-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a79cde7eb35c77bc49f45f4c927366a5be623ed091159f6b819b42f9236205f9","family_id":"qf-mod32-topology-interior","seed":202611012040,"semantic_template":"qf-mod32-topology-interior/{\"n\":5,\"subset\":[2,3],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,2,3],[0,3],[0]]}"},"fields":{"title":"Interior Unites All Open Subsets, Not One Convenient Open Set","stem":"Use the complete finite topology=[[], [0], [0, 3], [0, 2, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[2, 3]. Give (subset,open_subsets,interior), retaining original topology order among the qualifying opens.","answer":{"kind":"expression","value":"([2, 3], [[]], [])","explanation":"Select every open contained in the subset and take their union in increasing carrier order. The result is relative to this exact topology, not an unstated metric. The exact result, retaining the requested order, is ([2, 3], [[]], [])."},"skills":["Interior Unites All Open Subsets, Not One Convenient Open Set"],"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-20261101-r02-closure-and-interior-0041","batch_id":"batch-20261014-advance-20261101-r02-closure-and-interior-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"93e863efdc4b41df158a1d510ec8b043815f17a22e3b9d7897087bdaee693504","family_id":"qf-mod32-topology-interior","seed":202611012041,"semantic_template":"qf-mod32-topology-interior/{\"n\":5,\"subset\":[2,3],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,3],[0,3],[0]]}"},"fields":{"title":"Interior Unites All Open Subsets, Not One Convenient Open Set","stem":"Use the complete finite topology=[[], [0], [0, 3], [0, 2, 3], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[2, 3]. Give (subset,open_subsets,interior), retaining original topology order among the qualifying opens.","answer":{"kind":"expression","value":"([2, 3], [[]], [])","explanation":"Select every open contained in the subset and take their union in increasing carrier order. The result is relative to this exact topology, not an unstated metric. The exact result, retaining the requested order, is ([2, 3], [[]], [])."},"skills":["Interior Unites All Open Subsets, Not One Convenient Open Set"],"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-20261101-r02-closure-and-interior-0042","batch_id":"batch-20261014-advance-20261101-r02-closure-and-interior-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b0c437e4a762a4901617a22e835926da5f42bbdb512ce54432fcdc280d2d5e15","family_id":"qf-mod32-topology-interior","seed":202611012042,"semantic_template":"qf-mod32-topology-interior/{\"n\":5,\"subset\":[3,4],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,3,4],[0,3],[0]]}"},"fields":{"title":"Interior Unites All Open Subsets, Not One Convenient Open Set","stem":"Use the complete finite topology=[[], [0], [0, 3], [0, 3, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[3, 4]. Give (subset,open_subsets,interior), retaining original topology order among the qualifying opens.","answer":{"kind":"expression","value":"([3, 4], [[]], [])","explanation":"Select every open contained in the subset and take their union in increasing carrier order. The result is relative to this exact topology, not an unstated metric. The exact result, retaining the requested order, is ([3, 4], [[]], [])."},"skills":["Interior Unites All Open Subsets, Not One Convenient Open Set"],"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-20261101-r02-closure-and-interior-0043","batch_id":"batch-20261014-advance-20261101-r02-closure-and-interior-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"134df3898501c1264153e52345a757243701f0b4a5ce1484c54c554ada0dd069","family_id":"qf-mod32-topology-interior","seed":202611012043,"semantic_template":"qf-mod32-topology-interior/{\"n\":5,\"subset\":[3,4],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,3,4],[0,3],[0]]}"},"fields":{"title":"Interior Unites All Open Subsets, Not One Convenient Open Set","stem":"Use the complete finite topology=[[], [0], [0, 3], [0, 3, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[3, 4]. Give (subset,open_subsets,interior), retaining original topology order among the qualifying opens.","answer":{"kind":"expression","value":"([3, 4], [[]], [])","explanation":"Select every open contained in the subset and take their union in increasing carrier order. The result is relative to this exact topology, not an unstated metric. The exact result, retaining the requested order, is ([3, 4], [[]], [])."},"skills":["Interior Unites All Open Subsets, Not One Convenient Open Set"],"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-20261101-r02-closure-and-interior-0044","batch_id":"batch-20261014-advance-20261101-r02-closure-and-interior-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fef077fa391d7c0c41b526eefa129100e5d3db987bb80c8e25b071caba5bac38","family_id":"qf-mod32-topology-interior","seed":202611012044,"semantic_template":"qf-mod32-topology-interior/{\"n\":5,\"subset\":[1,4],\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,4],[0,4],[0]]}"},"fields":{"title":"Interior Unites All Open Subsets, Not One Convenient Open Set","stem":"Use the complete finite topology=[[], [0], [0, 4], [0, 1, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[1, 4]. Give (subset,open_subsets,interior), retaining original topology order among the qualifying opens.","answer":{"kind":"expression","value":"([1, 4], [[]], [])","explanation":"Select every open contained in the subset and take their union in increasing carrier order. The result is relative to this exact topology, not an unstated metric. The exact result, retaining the requested order, is ([1, 4], [[]], [])."},"skills":["Interior Unites All Open Subsets, Not One Convenient Open Set"],"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-20261101-r02-closure-and-interior-0045","batch_id":"batch-20261014-advance-20261101-r02-closure-and-interior-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"81f63a33d6b6404d7298b8bec6410949001a033dedb028d565e18520d2af3592","family_id":"qf-mod32-topology-interior","seed":202611012045,"semantic_template":"qf-mod32-topology-interior/{\"n\":5,\"subset\":[1,4],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,4],[0,4],[0]]}"},"fields":{"title":"Interior Unites All Open Subsets, Not One Convenient Open Set","stem":"Use the complete finite topology=[[], [0], [0, 4], [0, 1, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[1, 4]. Give (subset,open_subsets,interior), retaining original topology order among the qualifying opens.","answer":{"kind":"expression","value":"([1, 4], [[]], [])","explanation":"Select every open contained in the subset and take their union in increasing carrier order. The result is relative to this exact topology, not an unstated metric. The exact result, retaining the requested order, is ([1, 4], [[]], [])."},"skills":["Interior Unites All Open Subsets, Not One Convenient Open Set"],"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-20261101-r02-closure-and-interior-0046","batch_id":"batch-20261014-advance-20261101-r02-closure-and-interior-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8bf5447f25a8d317abaaee4a37c20772e712510a7bd93ec43258d8967b7f19d7","family_id":"qf-mod32-topology-interior","seed":202611012046,"semantic_template":"qf-mod32-topology-interior/{\"n\":5,\"subset\":[2,4],\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,2,4],[0,4],[0]]}"},"fields":{"title":"Interior Unites All Open Subsets, Not One Convenient Open Set","stem":"Use the complete finite topology=[[], [0], [0, 4], [0, 2, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[2, 4]. Give (subset,open_subsets,interior), retaining original topology order among the qualifying opens.","answer":{"kind":"expression","value":"([2, 4], [[]], [])","explanation":"Select every open contained in the subset and take their union in increasing carrier order. The result is relative to this exact topology, not an unstated metric. The exact result, retaining the requested order, is ([2, 4], [[]], [])."},"skills":["Interior Unites All Open Subsets, Not One Convenient Open Set"],"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-20261101-r02-closure-and-interior-0047","batch_id":"batch-20261014-advance-20261101-r02-closure-and-interior-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"55f2cd4c1868d808334ef6b7f673cacff56e854d26208aaec45bd71e391544a3","family_id":"qf-mod32-topology-interior","seed":202611012047,"semantic_template":"qf-mod32-topology-interior/{\"n\":5,\"subset\":[2,4],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,4],[0,4],[0]]}"},"fields":{"title":"Interior Unites All Open Subsets, Not One Convenient Open Set","stem":"Use the complete finite topology=[[], [0], [0, 4], [0, 2, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[2, 4]. Give (subset,open_subsets,interior), retaining original topology order among the qualifying opens.","answer":{"kind":"expression","value":"([2, 4], [[]], [])","explanation":"Select every open contained in the subset and take their union in increasing carrier order. The result is relative to this exact topology, not an unstated metric. The exact result, retaining the requested order, is ([2, 4], [[]], [])."},"skills":["Interior Unites All Open Subsets, Not One Convenient Open Set"],"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-20261101-r02-closure-and-interior-0048","batch_id":"batch-20261014-advance-20261101-r02-closure-and-interior-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0a9513199fd59ffbedad50fcac8928f92fab82cb13295943268e7fc47ee4b5f0","family_id":"qf-mod32-topology-interior","seed":202611012048,"semantic_template":"qf-mod32-topology-interior/{\"n\":5,\"subset\":[3,4],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,3,4],[0,4],[0]]}"},"fields":{"title":"Interior Unites All Open Subsets, Not One Convenient Open Set","stem":"Use the complete finite topology=[[], [0], [0, 4], [0, 3, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[3, 4]. Give (subset,open_subsets,interior), retaining original topology order among the qualifying opens.","answer":{"kind":"expression","value":"([3, 4], [[]], [])","explanation":"Select every open contained in the subset and take their union in increasing carrier order. The result is relative to this exact topology, not an unstated metric. The exact result, retaining the requested order, is ([3, 4], [[]], [])."},"skills":["Interior Unites All Open Subsets, Not One Convenient Open Set"],"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-20261101-r02-closure-and-interior-0049","batch_id":"batch-20261014-advance-20261101-r02-closure-and-interior-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c0bcf077c288c88b53e56013e08fc6260158c5cb21b796f03bbef12ba03a2c6a","family_id":"qf-mod32-topology-interior","seed":202611012049,"semantic_template":"qf-mod32-topology-interior/{\"n\":5,\"subset\":[3,4],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,3,4],[0,4],[0]]}"},"fields":{"title":"Interior Unites All Open Subsets, Not One Convenient Open Set","stem":"Use the complete finite topology=[[], [0], [0, 4], [0, 3, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[3, 4]. Give (subset,open_subsets,interior), retaining original topology order among the qualifying opens.","answer":{"kind":"expression","value":"([3, 4], [[]], [])","explanation":"Select every open contained in the subset and take their union in increasing carrier order. The result is relative to this exact topology, not an unstated metric. The exact result, retaining the requested order, is ([3, 4], [[]], [])."},"skills":["Interior Unites All Open Subsets, Not One Convenient Open Set"],"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-20261101-r02-closure-and-interior-0050","batch_id":"batch-20261014-advance-20261101-r02-closure-and-interior-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"027bc7fdd25edadb470e85da85e9278307764710bd835c6940db5785d4935132","family_id":"qf-mod32-topology-interior","seed":202611012050,"semantic_template":"qf-mod32-topology-interior/{\"n\":5,\"subset\":[0,2],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2],[0,1],[1]]}"},"fields":{"title":"Interior Unites All Open Subsets, Not One Convenient Open Set","stem":"Use the complete finite topology=[[], [1], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[0, 2]. Give (subset,open_subsets,interior), retaining original topology order among the qualifying opens.","answer":{"kind":"expression","value":"([0, 2], [[]], [])","explanation":"Select every open contained in the subset and take their union in increasing carrier order. The result is relative to this exact topology, not an unstated metric. The exact result, retaining the requested order, is ([0, 2], [[]], [])."},"skills":["Interior Unites All Open Subsets, Not One Convenient Open Set"],"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-20261101-r02-closure-and-interior-0051","batch_id":"batch-20261014-advance-20261101-r02-closure-and-interior-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bc758b2b7c57e29b8e02dc66486422808858de25f85494bc52b706bfaf1c8130","family_id":"qf-mod32-topology-closure-membership","seed":202611012051,"semantic_template":"qf-mod32-topology-closure-membership/{\"n\":5,\"point\":0,\"subset\":[1,2],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2],[0,1],[0]]}"},"fields":{"title":"Closure Membership May Hold outside the Original Subset","stem":"Use the complete finite topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[1, 2], point=0. Give (point,closure,in_closure?,topology_relative_closure).","answer":{"kind":"expression","value":"(0, [1, 2, 3, 4], No, topology_relative_closure)","explanation":"Compute the full intersection of closed supersets first, then test this point against that closure. Do not replace closure membership with membership in the original subset. The exact result, retaining the requested order, is (0, [1, 2, 3, 4], No, topology_relative_closure)."},"skills":["Closure Membership May Hold outside the Original Subset"],"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-20261101-r02-closure-and-interior-0052","batch_id":"batch-20261014-advance-20261101-r02-closure-and-interior-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eb1e039792591ce62c0a19187bb1c1ace541c25c8e763f319e3f197ef520dd82","family_id":"qf-mod32-topology-closure-membership","seed":202611012052,"semantic_template":"qf-mod32-topology-closure-membership/{\"n\":5,\"point\":1,\"subset\":[1,2],\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,2],[0,1],[0]]}"},"fields":{"title":"Closure Membership May Hold outside the Original Subset","stem":"Use the complete finite topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[1, 2], point=1. Give (point,closure,in_closure?,topology_relative_closure).","answer":{"kind":"expression","value":"(1, [1, 2, 3, 4], Yes, topology_relative_closure)","explanation":"Compute the full intersection of closed supersets first, then test this point against that closure. Do not replace closure membership with membership in the original subset. The exact result, retaining the requested order, is (1, [1, 2, 3, 4], Yes, topology_relative_closure)."},"skills":["Closure Membership May Hold outside the Original Subset"],"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-20261101-r02-closure-and-interior-0053","batch_id":"batch-20261014-advance-20261101-r02-closure-and-interior-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5c28af1a98ebd0b6d31e68dc99b66e26cf4c3c22ceeaaf1ba029bb0856377045","family_id":"qf-mod32-topology-closure-membership","seed":202611012053,"semantic_template":"qf-mod32-topology-closure-membership/{\"n\":5,\"point\":3,\"subset\":[1,3],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,3],[0,1],[0]]}"},"fields":{"title":"Closure Membership May Hold outside the Original Subset","stem":"Use the complete finite topology=[[], [0], [0, 1], [0, 1, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[1, 3], point=3. Give (point,closure,in_closure?,topology_relative_closure).","answer":{"kind":"expression","value":"(3, [1, 2, 3, 4], Yes, topology_relative_closure)","explanation":"Compute the full intersection of closed supersets first, then test this point against that closure. Do not replace closure membership with membership in the original subset. The exact result, retaining the requested order, is (3, [1, 2, 3, 4], Yes, topology_relative_closure)."},"skills":["Closure Membership May Hold outside the Original Subset"],"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-20261101-r02-closure-and-interior-0054","batch_id":"batch-20261014-advance-20261101-r02-closure-and-interior-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"02d0f8e0c1a7257927788c5e564a5d050905aebf084cf212c6e115d3a970e670","family_id":"qf-mod32-topology-closure-membership","seed":202611012054,"semantic_template":"qf-mod32-topology-closure-membership/{\"n\":5,\"point\":4,\"subset\":[1,3],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,3],[0,1],[0]]}"},"fields":{"title":"Closure Membership May Hold outside the Original Subset","stem":"Use the complete finite topology=[[], [0], [0, 1], [0, 1, 3], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[1, 3], point=4. Give (point,closure,in_closure?,topology_relative_closure).","answer":{"kind":"expression","value":"(4, [1, 2, 3, 4], Yes, topology_relative_closure)","explanation":"Compute the full intersection of closed supersets first, then test this point against that closure. Do not replace closure membership with membership in the original subset. The exact result, retaining the requested order, is (4, [1, 2, 3, 4], Yes, topology_relative_closure)."},"skills":["Closure Membership May Hold outside the Original Subset"],"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-20261101-r02-closure-and-interior-0055","batch_id":"batch-20261014-advance-20261101-r02-closure-and-interior-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aba918de903d9cbd79744d121626a557d8e59567f69888782c88815310d006d1","family_id":"qf-mod32-topology-closure-membership","seed":202611012055,"semantic_template":"qf-mod32-topology-closure-membership/{\"n\":5,\"point\":3,\"subset\":[1,4],\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,4],[0,1],[0]]}"},"fields":{"title":"Closure Membership May Hold outside the Original Subset","stem":"Use the complete finite topology=[[], [0], [0, 1], [0, 1, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[1, 4], point=3. Give (point,closure,in_closure?,topology_relative_closure).","answer":{"kind":"expression","value":"(3, [1, 2, 3, 4], Yes, topology_relative_closure)","explanation":"Compute the full intersection of closed supersets first, then test this point against that closure. Do not replace closure membership with membership in the original subset. The exact result, retaining the requested order, is (3, [1, 2, 3, 4], Yes, topology_relative_closure)."},"skills":["Closure Membership May Hold outside the Original Subset"],"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-20261101-r02-closure-and-interior-0056","batch_id":"batch-20261014-advance-20261101-r02-closure-and-interior-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7e336cc7e4ab442ddce2d0bb91fad74a88a9fc0ea93cfa7cfabdfc7e4a792669","family_id":"qf-mod32-topology-closure-membership","seed":202611012056,"semantic_template":"qf-mod32-topology-closure-membership/{\"n\":5,\"point\":0,\"subset\":[1,4],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,4],[0,1],[0]]}"},"fields":{"title":"Closure Membership May Hold outside the Original Subset","stem":"Use the complete finite topology=[[], [0], [0, 1], [0, 1, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[1, 4], point=0. Give (point,closure,in_closure?,topology_relative_closure).","answer":{"kind":"expression","value":"(0, [1, 2, 3, 4], No, topology_relative_closure)","explanation":"Compute the full intersection of closed supersets first, then test this point against that closure. Do not replace closure membership with membership in the original subset. The exact result, retaining the requested order, is (0, [1, 2, 3, 4], No, topology_relative_closure)."},"skills":["Closure Membership May Hold outside the Original Subset"],"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-20261101-r02-closure-and-interior-0057","batch_id":"batch-20261014-advance-20261101-r02-closure-and-interior-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"182e8fdabf91eba48e978c75632d3a25a6122e9f07036ffa11f07793a1ab5be3","family_id":"qf-mod32-topology-closure-membership","seed":202611012057,"semantic_template":"qf-mod32-topology-closure-membership/{\"n\":5,\"point\":2,\"subset\":[1,2],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2],[0,2],[0]]}"},"fields":{"title":"Closure Membership May Hold outside the Original Subset","stem":"Use the complete finite topology=[[], [0], [0, 2], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[1, 2], point=2. Give (point,closure,in_closure?,topology_relative_closure).","answer":{"kind":"expression","value":"(2, [1, 2, 3, 4], Yes, topology_relative_closure)","explanation":"Compute the full intersection of closed supersets first, then test this point against that closure. Do not replace closure membership with membership in the original subset. The exact result, retaining the requested order, is (2, [1, 2, 3, 4], Yes, topology_relative_closure)."},"skills":["Closure Membership May Hold outside the Original Subset"],"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-20261101-r02-closure-and-interior-0058","batch_id":"batch-20261014-advance-20261101-r02-closure-and-interior-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a9b782a3ae5659cf0476ac153b8ad8d5a59042c47930657f549e136bbc02a694","family_id":"qf-mod32-topology-closure-membership","seed":202611012058,"semantic_template":"qf-mod32-topology-closure-membership/{\"n\":5,\"point\":1,\"subset\":[1,2],\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,2],[0,2],[0]]}"},"fields":{"title":"Closure Membership May Hold outside the Original Subset","stem":"Use the complete finite topology=[[], [0], [0, 2], [0, 1, 2], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[1, 2], point=1. Give (point,closure,in_closure?,topology_relative_closure).","answer":{"kind":"expression","value":"(1, [1, 2, 3, 4], Yes, topology_relative_closure)","explanation":"Compute the full intersection of closed supersets first, then test this point against that closure. Do not replace closure membership with membership in the original subset. The exact result, retaining the requested order, is (1, [1, 2, 3, 4], Yes, topology_relative_closure)."},"skills":["Closure Membership May Hold outside the Original Subset"],"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-20261101-r02-closure-and-interior-0059","batch_id":"batch-20261014-advance-20261101-r02-closure-and-interior-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a5436743945dedb6fceae866c04c2d267e618ebbf355baba37fc69ae644463c9","family_id":"qf-mod32-topology-closure-membership","seed":202611012059,"semantic_template":"qf-mod32-topology-closure-membership/{\"n\":5,\"point\":1,\"subset\":[2,3],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,2,3],[0,2],[0]]}"},"fields":{"title":"Closure Membership May Hold outside the Original Subset","stem":"Use the complete finite topology=[[], [0], [0, 2], [0, 2, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[2, 3], point=1. Give (point,closure,in_closure?,topology_relative_closure).","answer":{"kind":"expression","value":"(1, [1, 2, 3, 4], Yes, topology_relative_closure)","explanation":"Compute the full intersection of closed supersets first, then test this point against that closure. Do not replace closure membership with membership in the original subset. The exact result, retaining the requested order, is (1, [1, 2, 3, 4], Yes, topology_relative_closure)."},"skills":["Closure Membership May Hold outside the Original Subset"],"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-20261101-r02-closure-and-interior-0060","batch_id":"batch-20261014-advance-20261101-r02-closure-and-interior-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c1e6c2f979369808ffdad608cb56df08186259a1da635ae6965009911b643f0b","family_id":"qf-mod32-topology-closure-membership","seed":202611012060,"semantic_template":"qf-mod32-topology-closure-membership/{\"n\":5,\"point\":1,\"subset\":[2,3],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,3],[0,2],[0]]}"},"fields":{"title":"Closure Membership May Hold outside the Original Subset","stem":"Use the complete finite topology=[[], [0], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[2, 3], point=1. Give (point,closure,in_closure?,topology_relative_closure).","answer":{"kind":"expression","value":"(1, [1, 2, 3, 4], Yes, topology_relative_closure)","explanation":"Compute the full intersection of closed supersets first, then test this point against that closure. Do not replace closure membership with membership in the original subset. The exact result, retaining the requested order, is (1, [1, 2, 3, 4], Yes, topology_relative_closure)."},"skills":["Closure Membership May Hold outside the Original Subset"],"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-20261101-r02-closure-and-interior-0061","batch_id":"batch-20261014-advance-20261101-r02-closure-and-interior-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8d236bce2a5718966530a006799d242cd16741d8127381764decbe664525d7f8","family_id":"qf-mod32-topology-closure-membership","seed":202611012061,"semantic_template":"qf-mod32-topology-closure-membership/{\"n\":5,\"point\":0,\"subset\":[2,4],\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,2,4],[0,2],[0]]}"},"fields":{"title":"Closure Membership May Hold outside the Original Subset","stem":"Use the complete finite topology=[[], [0], [0, 2], [0, 2, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[2, 4], point=0. Give (point,closure,in_closure?,topology_relative_closure).","answer":{"kind":"expression","value":"(0, [1, 2, 3, 4], No, topology_relative_closure)","explanation":"Compute the full intersection of closed supersets first, then test this point against that closure. Do not replace closure membership with membership in the original subset. The exact result, retaining the requested order, is (0, [1, 2, 3, 4], No, topology_relative_closure)."},"skills":["Closure Membership May Hold outside the Original Subset"],"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-20261101-r02-closure-and-interior-0062","batch_id":"batch-20261014-advance-20261101-r02-closure-and-interior-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a9eae52186abe9c25955c251a0f7334b2d1bd890daa4fd1b6cc2c6f51ec50661","family_id":"qf-mod32-topology-closure-membership","seed":202611012062,"semantic_template":"qf-mod32-topology-closure-membership/{\"n\":5,\"point\":2,\"subset\":[2,4],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,4],[0,2],[0]]}"},"fields":{"title":"Closure Membership May Hold outside the Original Subset","stem":"Use the complete finite topology=[[], [0], [0, 2], [0, 2, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[2, 4], point=2. Give (point,closure,in_closure?,topology_relative_closure).","answer":{"kind":"expression","value":"(2, [1, 2, 3, 4], Yes, topology_relative_closure)","explanation":"Compute the full intersection of closed supersets first, then test this point against that closure. Do not replace closure membership with membership in the original subset. The exact result, retaining the requested order, is (2, [1, 2, 3, 4], Yes, topology_relative_closure)."},"skills":["Closure Membership May Hold outside the Original Subset"],"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-20261101-r02-closure-and-interior-0063","batch_id":"batch-20261014-advance-20261101-r02-closure-and-interior-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c076647457885d4e4c2f304eeb33f631eb7dcb242e1ba49b97ce93edba8968dc","family_id":"qf-mod32-topology-closure-membership","seed":202611012063,"semantic_template":"qf-mod32-topology-closure-membership/{\"n\":5,\"point\":1,\"subset\":[1,3],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,3],[0,3],[0]]}"},"fields":{"title":"Closure Membership May Hold outside the Original Subset","stem":"Use the complete finite topology=[[], [0], [0, 3], [0, 1, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[1, 3], point=1. Give (point,closure,in_closure?,topology_relative_closure).","answer":{"kind":"expression","value":"(1, [1, 2, 3, 4], Yes, topology_relative_closure)","explanation":"Compute the full intersection of closed supersets first, then test this point against that closure. Do not replace closure membership with membership in the original subset. The exact result, retaining the requested order, is (1, [1, 2, 3, 4], Yes, topology_relative_closure)."},"skills":["Closure Membership May Hold outside the Original Subset"],"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-20261101-r02-closure-and-interior-0064","batch_id":"batch-20261014-advance-20261101-r02-closure-and-interior-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"439ca65ac68485e78bd3a87a4ae0c052915193882a511feec4729018f3b4dd8d","family_id":"qf-mod32-topology-closure-membership","seed":202611012064,"semantic_template":"qf-mod32-topology-closure-membership/{\"n\":5,\"point\":4,\"subset\":[1,3],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,3],[0,3],[0]]}"},"fields":{"title":"Closure Membership May Hold outside the Original Subset","stem":"Use the complete finite topology=[[], [0], [0, 3], [0, 1, 3], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[1, 3], point=4. Give (point,closure,in_closure?,topology_relative_closure).","answer":{"kind":"expression","value":"(4, [1, 2, 3, 4], Yes, topology_relative_closure)","explanation":"Compute the full intersection of closed supersets first, then test this point against that closure. Do not replace closure membership with membership in the original subset. The exact result, retaining the requested order, is (4, [1, 2, 3, 4], Yes, topology_relative_closure)."},"skills":["Closure Membership May Hold outside the Original Subset"],"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-20261101-r02-closure-and-interior-0065","batch_id":"batch-20261014-advance-20261101-r02-closure-and-interior-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"492b3246a4874975241d887a0319938a1ea5bb68facf08db7ab8078c12cffe13","family_id":"qf-mod32-topology-closure-membership","seed":202611012065,"semantic_template":"qf-mod32-topology-closure-membership/{\"n\":5,\"point\":4,\"subset\":[2,3],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,2,3],[0,3],[0]]}"},"fields":{"title":"Closure Membership May Hold outside the Original Subset","stem":"Use the complete finite topology=[[], [0], [0, 3], [0, 2, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[2, 3], point=4. Give (point,closure,in_closure?,topology_relative_closure).","answer":{"kind":"expression","value":"(4, [1, 2, 3, 4], Yes, topology_relative_closure)","explanation":"Compute the full intersection of closed supersets first, then test this point against that closure. Do not replace closure membership with membership in the original subset. The exact result, retaining the requested order, is (4, [1, 2, 3, 4], Yes, topology_relative_closure)."},"skills":["Closure Membership May Hold outside the Original Subset"],"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-20261101-r02-closure-and-interior-0066","batch_id":"batch-20261014-advance-20261101-r02-closure-and-interior-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9902f6469e77f7c2b53618d6cd288af0f837382e4eaa04875e2fb566b82ed0fc","family_id":"qf-mod32-topology-closure-membership","seed":202611012066,"semantic_template":"qf-mod32-topology-closure-membership/{\"n\":5,\"point\":0,\"subset\":[2,3],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,3],[0,3],[0]]}"},"fields":{"title":"Closure Membership May Hold outside the Original Subset","stem":"Use the complete finite topology=[[], [0], [0, 3], [0, 2, 3], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[2, 3], point=0. Give (point,closure,in_closure?,topology_relative_closure).","answer":{"kind":"expression","value":"(0, [1, 2, 3, 4], No, topology_relative_closure)","explanation":"Compute the full intersection of closed supersets first, then test this point against that closure. Do not replace closure membership with membership in the original subset. The exact result, retaining the requested order, is (0, [1, 2, 3, 4], No, topology_relative_closure)."},"skills":["Closure Membership May Hold outside the Original Subset"],"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-20261101-r02-closure-and-interior-0067","batch_id":"batch-20261014-advance-20261101-r02-closure-and-interior-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a1d467c20b37273009666361062d7d59017f54595890778baae5d7e0cfeaa227","family_id":"qf-mod32-topology-closure-membership","seed":202611012067,"semantic_template":"qf-mod32-topology-closure-membership/{\"n\":5,\"point\":3,\"subset\":[3,4],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,3,4],[0,3],[0]]}"},"fields":{"title":"Closure Membership May Hold outside the Original Subset","stem":"Use the complete finite topology=[[], [0], [0, 3], [0, 3, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[3, 4], point=3. Give (point,closure,in_closure?,topology_relative_closure).","answer":{"kind":"expression","value":"(3, [1, 2, 3, 4], Yes, topology_relative_closure)","explanation":"Compute the full intersection of closed supersets first, then test this point against that closure. Do not replace closure membership with membership in the original subset. The exact result, retaining the requested order, is (3, [1, 2, 3, 4], Yes, topology_relative_closure)."},"skills":["Closure Membership May Hold outside the Original Subset"],"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-20261101-r02-closure-and-interior-0068","batch_id":"batch-20261014-advance-20261101-r02-closure-and-interior-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7db34e7170fe86e5f148cdd6e0302445fa678a71ca187580b10fae5abab4c1b4","family_id":"qf-mod32-topology-closure-membership","seed":202611012068,"semantic_template":"qf-mod32-topology-closure-membership/{\"n\":5,\"point\":4,\"subset\":[3,4],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,3,4],[0,3],[0]]}"},"fields":{"title":"Closure Membership May Hold outside the Original Subset","stem":"Use the complete finite topology=[[], [0], [0, 3], [0, 3, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[3, 4], point=4. Give (point,closure,in_closure?,topology_relative_closure).","answer":{"kind":"expression","value":"(4, [1, 2, 3, 4], Yes, topology_relative_closure)","explanation":"Compute the full intersection of closed supersets first, then test this point against that closure. Do not replace closure membership with membership in the original subset. The exact result, retaining the requested order, is (4, [1, 2, 3, 4], Yes, topology_relative_closure)."},"skills":["Closure Membership May Hold outside the Original Subset"],"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-20261101-r02-closure-and-interior-0069","batch_id":"batch-20261014-advance-20261101-r02-closure-and-interior-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"55f9b9f3740d05b2c3e149fbcc3f47defe248607c75680dca99ab11e193ecb36","family_id":"qf-mod32-topology-closure-membership","seed":202611012069,"semantic_template":"qf-mod32-topology-closure-membership/{\"n\":5,\"point\":2,\"subset\":[1,4],\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,4],[0,4],[0]]}"},"fields":{"title":"Closure Membership May Hold outside the Original Subset","stem":"Use the complete finite topology=[[], [0], [0, 4], [0, 1, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[1, 4], point=2. Give (point,closure,in_closure?,topology_relative_closure).","answer":{"kind":"expression","value":"(2, [1, 2, 3, 4], Yes, topology_relative_closure)","explanation":"Compute the full intersection of closed supersets first, then test this point against that closure. Do not replace closure membership with membership in the original subset. The exact result, retaining the requested order, is (2, [1, 2, 3, 4], Yes, topology_relative_closure)."},"skills":["Closure Membership May Hold outside the Original Subset"],"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-20261101-r02-closure-and-interior-0070","batch_id":"batch-20261014-advance-20261101-r02-closure-and-interior-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dab09c7af1febed72d51b538cb6c99a4d7ef9e4c70527ab7f31ada3a215eb60b","family_id":"qf-mod32-topology-closure-membership","seed":202611012070,"semantic_template":"qf-mod32-topology-closure-membership/{\"n\":5,\"point\":2,\"subset\":[1,4],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,4],[0,4],[0]]}"},"fields":{"title":"Closure Membership May Hold outside the Original Subset","stem":"Use the complete finite topology=[[], [0], [0, 4], [0, 1, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[1, 4], point=2. Give (point,closure,in_closure?,topology_relative_closure).","answer":{"kind":"expression","value":"(2, [1, 2, 3, 4], Yes, topology_relative_closure)","explanation":"Compute the full intersection of closed supersets first, then test this point against that closure. Do not replace closure membership with membership in the original subset. The exact result, retaining the requested order, is (2, [1, 2, 3, 4], Yes, topology_relative_closure)."},"skills":["Closure Membership May Hold outside the Original Subset"],"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-20261101-r02-closure-and-interior-0071","batch_id":"batch-20261014-advance-20261101-r02-closure-and-interior-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"75a843b302c2efa6a80e3434c1502999def34293200a2c2eb8f402403bf5e699","family_id":"qf-mod32-topology-closure-membership","seed":202611012071,"semantic_template":"qf-mod32-topology-closure-membership/{\"n\":5,\"point\":0,\"subset\":[2,4],\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,2,4],[0,4],[0]]}"},"fields":{"title":"Closure Membership May Hold outside the Original Subset","stem":"Use the complete finite topology=[[], [0], [0, 4], [0, 2, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[2, 4], point=0. Give (point,closure,in_closure?,topology_relative_closure).","answer":{"kind":"expression","value":"(0, [1, 2, 3, 4], No, topology_relative_closure)","explanation":"Compute the full intersection of closed supersets first, then test this point against that closure. Do not replace closure membership with membership in the original subset. The exact result, retaining the requested order, is (0, [1, 2, 3, 4], No, topology_relative_closure)."},"skills":["Closure Membership May Hold outside the Original Subset"],"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-20261101-r02-closure-and-interior-0072","batch_id":"batch-20261014-advance-20261101-r02-closure-and-interior-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1134199938eb069da13e9d8a2a5924f3860637a8f7222b01bb2a03127e864fb3","family_id":"qf-mod32-topology-closure-membership","seed":202611012072,"semantic_template":"qf-mod32-topology-closure-membership/{\"n\":5,\"point\":4,\"subset\":[2,4],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,4],[0,4],[0]]}"},"fields":{"title":"Closure Membership May Hold outside the Original Subset","stem":"Use the complete finite topology=[[], [0], [0, 4], [0, 2, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[2, 4], point=4. Give (point,closure,in_closure?,topology_relative_closure).","answer":{"kind":"expression","value":"(4, [1, 2, 3, 4], Yes, topology_relative_closure)","explanation":"Compute the full intersection of closed supersets first, then test this point against that closure. Do not replace closure membership with membership in the original subset. The exact result, retaining the requested order, is (4, [1, 2, 3, 4], Yes, topology_relative_closure)."},"skills":["Closure Membership May Hold outside the Original Subset"],"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-20261101-r02-closure-and-interior-0073","batch_id":"batch-20261014-advance-20261101-r02-closure-and-interior-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a768aadce68aa84969e8f48d88c88d188e4757220c2f264e21f7f86ce8468e0c","family_id":"qf-mod32-topology-closure-membership","seed":202611012073,"semantic_template":"qf-mod32-topology-closure-membership/{\"n\":5,\"point\":3,\"subset\":[3,4],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,3,4],[0,4],[0]]}"},"fields":{"title":"Closure Membership May Hold outside the Original Subset","stem":"Use the complete finite topology=[[], [0], [0, 4], [0, 3, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[3, 4], point=3. Give (point,closure,in_closure?,topology_relative_closure).","answer":{"kind":"expression","value":"(3, [1, 2, 3, 4], Yes, topology_relative_closure)","explanation":"Compute the full intersection of closed supersets first, then test this point against that closure. Do not replace closure membership with membership in the original subset. The exact result, retaining the requested order, is (3, [1, 2, 3, 4], Yes, topology_relative_closure)."},"skills":["Closure Membership May Hold outside the Original Subset"],"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-20261101-r02-closure-and-interior-0074","batch_id":"batch-20261014-advance-20261101-r02-closure-and-interior-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a97335b54824aff993f9b890e7696ae0bb6d371bf37007fb87de6397ed683e7a","family_id":"qf-mod32-topology-closure-membership","seed":202611012074,"semantic_template":"qf-mod32-topology-closure-membership/{\"n\":5,\"point\":2,\"subset\":[3,4],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,3,4],[0,4],[0]]}"},"fields":{"title":"Closure Membership May Hold outside the Original Subset","stem":"Use the complete finite topology=[[], [0], [0, 4], [0, 3, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[3, 4], point=2. Give (point,closure,in_closure?,topology_relative_closure).","answer":{"kind":"expression","value":"(2, [1, 2, 3, 4], Yes, topology_relative_closure)","explanation":"Compute the full intersection of closed supersets first, then test this point against that closure. Do not replace closure membership with membership in the original subset. The exact result, retaining the requested order, is (2, [1, 2, 3, 4], Yes, topology_relative_closure)."},"skills":["Closure Membership May Hold outside the Original Subset"],"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-20261101-r02-closure-and-interior-0075","batch_id":"batch-20261014-advance-20261101-r02-closure-and-interior-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4788840d5730c92070137b23a12d93e0383c52c7e5225b5b17e15388e9e46f1f","family_id":"qf-mod32-topology-closure-membership","seed":202611012075,"semantic_template":"qf-mod32-topology-closure-membership/{\"n\":5,\"point\":4,\"subset\":[0,2],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2],[0,1],[1]]}"},"fields":{"title":"Closure Membership May Hold outside the Original Subset","stem":"Use the complete finite topology=[[], [1], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[0, 2], point=4. Give (point,closure,in_closure?,topology_relative_closure).","answer":{"kind":"expression","value":"(4, [0, 2, 3, 4], Yes, topology_relative_closure)","explanation":"Compute the full intersection of closed supersets first, then test this point against that closure. Do not replace closure membership with membership in the original subset. The exact result, retaining the requested order, is (4, [0, 2, 3, 4], Yes, topology_relative_closure)."},"skills":["Closure Membership May Hold outside the Original Subset"],"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-20261101-r02-closure-and-interior-0076","batch_id":"batch-20261014-advance-20261101-r02-closure-and-interior-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"64e8d7ca19a4117c0d41bf81b9260f967c02f2d95f9e46c413b2527c41450779","family_id":"qf-mod32-topology-interior-membership","seed":202611012076,"semantic_template":"qf-mod32-topology-interior-membership/{\"n\":5,\"point\":0,\"subset\":[1,2],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2],[0,1],[0]]}"},"fields":{"title":"Interior Membership Needs an Open Neighborhood inside the Subset","stem":"Use the complete finite topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[1, 2], point=0. Give (point,interior,in_interior?,topology_relative_interior).","answer":{"kind":"expression","value":"(0, [], No, topology_relative_interior)","explanation":"Unite all opens contained in the subset, then test the point. Membership in the subset alone need not give membership in its interior. The exact result, retaining the requested order, is (0, [], No, topology_relative_interior)."},"skills":["Interior Membership Needs an Open Neighborhood inside the Subset"],"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-20261101-r02-closure-and-interior-0077","batch_id":"batch-20261014-advance-20261101-r02-closure-and-interior-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d13a79d093470974024fbfca71cda03f977495271577206ceed8eaf98ee0a00f","family_id":"qf-mod32-topology-interior-membership","seed":202611012077,"semantic_template":"qf-mod32-topology-interior-membership/{\"n\":5,\"point\":1,\"subset\":[1,2],\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,2],[0,1],[0]]}"},"fields":{"title":"Interior Membership Needs an Open Neighborhood inside the Subset","stem":"Use the complete finite topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[1, 2], point=1. Give (point,interior,in_interior?,topology_relative_interior).","answer":{"kind":"expression","value":"(1, [], No, topology_relative_interior)","explanation":"Unite all opens contained in the subset, then test the point. Membership in the subset alone need not give membership in its interior. The exact result, retaining the requested order, is (1, [], No, topology_relative_interior)."},"skills":["Interior Membership Needs an Open Neighborhood inside the Subset"],"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-20261101-r02-closure-and-interior-0078","batch_id":"batch-20261014-advance-20261101-r02-closure-and-interior-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cdbd6f8674f5b4f2e53f96d059c7126a33bc36abef8be79946bde23d5b7d1047","family_id":"qf-mod32-topology-interior-membership","seed":202611012078,"semantic_template":"qf-mod32-topology-interior-membership/{\"n\":5,\"point\":3,\"subset\":[1,3],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,3],[0,1],[0]]}"},"fields":{"title":"Interior Membership Needs an Open Neighborhood inside the Subset","stem":"Use the complete finite topology=[[], [0], [0, 1], [0, 1, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[1, 3], point=3. Give (point,interior,in_interior?,topology_relative_interior).","answer":{"kind":"expression","value":"(3, [], No, topology_relative_interior)","explanation":"Unite all opens contained in the subset, then test the point. Membership in the subset alone need not give membership in its interior. The exact result, retaining the requested order, is (3, [], No, topology_relative_interior)."},"skills":["Interior Membership Needs an Open Neighborhood inside the Subset"],"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-20261101-r02-closure-and-interior-0079","batch_id":"batch-20261014-advance-20261101-r02-closure-and-interior-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9294fa13be6e86cafb1389126da9175283d691babc065070b538f5f0b2a12bb4","family_id":"qf-mod32-topology-interior-membership","seed":202611012079,"semantic_template":"qf-mod32-topology-interior-membership/{\"n\":5,\"point\":4,\"subset\":[1,3],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,3],[0,1],[0]]}"},"fields":{"title":"Interior Membership Needs an Open Neighborhood inside the Subset","stem":"Use the complete finite topology=[[], [0], [0, 1], [0, 1, 3], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[1, 3], point=4. Give (point,interior,in_interior?,topology_relative_interior).","answer":{"kind":"expression","value":"(4, [], No, topology_relative_interior)","explanation":"Unite all opens contained in the subset, then test the point. Membership in the subset alone need not give membership in its interior. The exact result, retaining the requested order, is (4, [], No, topology_relative_interior)."},"skills":["Interior Membership Needs an Open Neighborhood inside the Subset"],"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-20261101-r02-closure-and-interior-0080","batch_id":"batch-20261014-advance-20261101-r02-closure-and-interior-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"24bfba35fa52dfeb6ef7518ade731884c1a218e9a99c008df246a5b8ec2d4881","family_id":"qf-mod32-topology-interior-membership","seed":202611012080,"semantic_template":"qf-mod32-topology-interior-membership/{\"n\":5,\"point\":3,\"subset\":[1,4],\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,4],[0,1],[0]]}"},"fields":{"title":"Interior Membership Needs an Open Neighborhood inside the Subset","stem":"Use the complete finite topology=[[], [0], [0, 1], [0, 1, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[1, 4], point=3. Give (point,interior,in_interior?,topology_relative_interior).","answer":{"kind":"expression","value":"(3, [], No, topology_relative_interior)","explanation":"Unite all opens contained in the subset, then test the point. Membership in the subset alone need not give membership in its interior. The exact result, retaining the requested order, is (3, [], No, topology_relative_interior)."},"skills":["Interior Membership Needs an Open Neighborhood inside the Subset"],"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-20261101-r02-closure-and-interior-0081","batch_id":"batch-20261014-advance-20261101-r02-closure-and-interior-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d7d24281a1a21484f8fac0bca3fa81447724de7a239c19cb85c15681c66c2543","family_id":"qf-mod32-topology-interior-membership","seed":202611012081,"semantic_template":"qf-mod32-topology-interior-membership/{\"n\":5,\"point\":0,\"subset\":[1,4],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,4],[0,1],[0]]}"},"fields":{"title":"Interior Membership Needs an Open Neighborhood inside the Subset","stem":"Use the complete finite topology=[[], [0], [0, 1], [0, 1, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[1, 4], point=0. Give (point,interior,in_interior?,topology_relative_interior).","answer":{"kind":"expression","value":"(0, [], No, topology_relative_interior)","explanation":"Unite all opens contained in the subset, then test the point. Membership in the subset alone need not give membership in its interior. The exact result, retaining the requested order, is (0, [], No, topology_relative_interior)."},"skills":["Interior Membership Needs an Open Neighborhood inside the Subset"],"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-20261101-r02-closure-and-interior-0082","batch_id":"batch-20261014-advance-20261101-r02-closure-and-interior-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f534527f78a1ed72e396c105147ed68a27ea4911e7268cf16ed5c1249aabcd64","family_id":"qf-mod32-topology-interior-membership","seed":202611012082,"semantic_template":"qf-mod32-topology-interior-membership/{\"n\":5,\"point\":2,\"subset\":[1,2],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2],[0,2],[0]]}"},"fields":{"title":"Interior Membership Needs an Open Neighborhood inside the Subset","stem":"Use the complete finite topology=[[], [0], [0, 2], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[1, 2], point=2. Give (point,interior,in_interior?,topology_relative_interior).","answer":{"kind":"expression","value":"(2, [], No, topology_relative_interior)","explanation":"Unite all opens contained in the subset, then test the point. Membership in the subset alone need not give membership in its interior. The exact result, retaining the requested order, is (2, [], No, topology_relative_interior)."},"skills":["Interior Membership Needs an Open Neighborhood inside the Subset"],"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-20261101-r02-closure-and-interior-0083","batch_id":"batch-20261014-advance-20261101-r02-closure-and-interior-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8c38eea78e3472d5de923930351d29566c3095f0424a7e18453bed220f71672a","family_id":"qf-mod32-topology-interior-membership","seed":202611012083,"semantic_template":"qf-mod32-topology-interior-membership/{\"n\":5,\"point\":1,\"subset\":[1,2],\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,2],[0,2],[0]]}"},"fields":{"title":"Interior Membership Needs an Open Neighborhood inside the Subset","stem":"Use the complete finite topology=[[], [0], [0, 2], [0, 1, 2], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[1, 2], point=1. Give (point,interior,in_interior?,topology_relative_interior).","answer":{"kind":"expression","value":"(1, [], No, topology_relative_interior)","explanation":"Unite all opens contained in the subset, then test the point. Membership in the subset alone need not give membership in its interior. The exact result, retaining the requested order, is (1, [], No, topology_relative_interior)."},"skills":["Interior Membership Needs an Open Neighborhood inside the Subset"],"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-20261101-r02-closure-and-interior-0084","batch_id":"batch-20261014-advance-20261101-r02-closure-and-interior-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a8d05cdc0ffbf0dc2123c69f1c76357171071d2a280ef71fad640975558e27c3","family_id":"qf-mod32-topology-interior-membership","seed":202611012084,"semantic_template":"qf-mod32-topology-interior-membership/{\"n\":5,\"point\":1,\"subset\":[2,3],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,2,3],[0,2],[0]]}"},"fields":{"title":"Interior Membership Needs an Open Neighborhood inside the Subset","stem":"Use the complete finite topology=[[], [0], [0, 2], [0, 2, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[2, 3], point=1. Give (point,interior,in_interior?,topology_relative_interior).","answer":{"kind":"expression","value":"(1, [], No, topology_relative_interior)","explanation":"Unite all opens contained in the subset, then test the point. Membership in the subset alone need not give membership in its interior. The exact result, retaining the requested order, is (1, [], No, topology_relative_interior)."},"skills":["Interior Membership Needs an Open Neighborhood inside the Subset"],"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-20261101-r02-closure-and-interior-0085","batch_id":"batch-20261014-advance-20261101-r02-closure-and-interior-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"984a8295c058cca9e5283d9a3de536179d9a5885b4e5675ca907329f2fac35a7","family_id":"qf-mod32-topology-interior-membership","seed":202611012085,"semantic_template":"qf-mod32-topology-interior-membership/{\"n\":5,\"point\":1,\"subset\":[2,3],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,3],[0,2],[0]]}"},"fields":{"title":"Interior Membership Needs an Open Neighborhood inside the Subset","stem":"Use the complete finite topology=[[], [0], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[2, 3], point=1. Give (point,interior,in_interior?,topology_relative_interior).","answer":{"kind":"expression","value":"(1, [], No, topology_relative_interior)","explanation":"Unite all opens contained in the subset, then test the point. Membership in the subset alone need not give membership in its interior. The exact result, retaining the requested order, is (1, [], No, topology_relative_interior)."},"skills":["Interior Membership Needs an Open Neighborhood inside the Subset"],"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-20261101-r02-closure-and-interior-0086","batch_id":"batch-20261014-advance-20261101-r02-closure-and-interior-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cc87b78e4111f99a0174f3f91764d9ac7f460661f53b04f47156e19c74ad53fd","family_id":"qf-mod32-topology-interior-membership","seed":202611012086,"semantic_template":"qf-mod32-topology-interior-membership/{\"n\":5,\"point\":0,\"subset\":[2,4],\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,2,4],[0,2],[0]]}"},"fields":{"title":"Interior Membership Needs an Open Neighborhood inside the Subset","stem":"Use the complete finite topology=[[], [0], [0, 2], [0, 2, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[2, 4], point=0. Give (point,interior,in_interior?,topology_relative_interior).","answer":{"kind":"expression","value":"(0, [], No, topology_relative_interior)","explanation":"Unite all opens contained in the subset, then test the point. Membership in the subset alone need not give membership in its interior. The exact result, retaining the requested order, is (0, [], No, topology_relative_interior)."},"skills":["Interior Membership Needs an Open Neighborhood inside the Subset"],"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-20261101-r02-closure-and-interior-0087","batch_id":"batch-20261014-advance-20261101-r02-closure-and-interior-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d811c0d9aeae906bf8d9e3df5817ecea693dfbe32289f1d4c571de8d7e17f7db","family_id":"qf-mod32-topology-interior-membership","seed":202611012087,"semantic_template":"qf-mod32-topology-interior-membership/{\"n\":5,\"point\":2,\"subset\":[2,4],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,4],[0,2],[0]]}"},"fields":{"title":"Interior Membership Needs an Open Neighborhood inside the Subset","stem":"Use the complete finite topology=[[], [0], [0, 2], [0, 2, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[2, 4], point=2. Give (point,interior,in_interior?,topology_relative_interior).","answer":{"kind":"expression","value":"(2, [], No, topology_relative_interior)","explanation":"Unite all opens contained in the subset, then test the point. Membership in the subset alone need not give membership in its interior. The exact result, retaining the requested order, is (2, [], No, topology_relative_interior)."},"skills":["Interior Membership Needs an Open Neighborhood inside the Subset"],"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-20261101-r02-closure-and-interior-0088","batch_id":"batch-20261014-advance-20261101-r02-closure-and-interior-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4b83eb0ed5c666f6484bf87ddf6aceb2491c13d5d4b3d719cd439d3be7bc268d","family_id":"qf-mod32-topology-interior-membership","seed":202611012088,"semantic_template":"qf-mod32-topology-interior-membership/{\"n\":5,\"point\":1,\"subset\":[1,3],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,3],[0,3],[0]]}"},"fields":{"title":"Interior Membership Needs an Open Neighborhood inside the Subset","stem":"Use the complete finite topology=[[], [0], [0, 3], [0, 1, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[1, 3], point=1. Give (point,interior,in_interior?,topology_relative_interior).","answer":{"kind":"expression","value":"(1, [], No, topology_relative_interior)","explanation":"Unite all opens contained in the subset, then test the point. Membership in the subset alone need not give membership in its interior. The exact result, retaining the requested order, is (1, [], No, topology_relative_interior)."},"skills":["Interior Membership Needs an Open Neighborhood inside the Subset"],"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-20261101-r02-closure-and-interior-0089","batch_id":"batch-20261014-advance-20261101-r02-closure-and-interior-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"36b06b8eaec3e30da1975ea08df7a523f6177220433e683a046b42dc8a5d7435","family_id":"qf-mod32-topology-interior-membership","seed":202611012089,"semantic_template":"qf-mod32-topology-interior-membership/{\"n\":5,\"point\":4,\"subset\":[1,3],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,3],[0,3],[0]]}"},"fields":{"title":"Interior Membership Needs an Open Neighborhood inside the Subset","stem":"Use the complete finite topology=[[], [0], [0, 3], [0, 1, 3], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[1, 3], point=4. Give (point,interior,in_interior?,topology_relative_interior).","answer":{"kind":"expression","value":"(4, [], No, topology_relative_interior)","explanation":"Unite all opens contained in the subset, then test the point. Membership in the subset alone need not give membership in its interior. The exact result, retaining the requested order, is (4, [], No, topology_relative_interior)."},"skills":["Interior Membership Needs an Open Neighborhood inside the Subset"],"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-20261101-r02-closure-and-interior-0090","batch_id":"batch-20261014-advance-20261101-r02-closure-and-interior-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e4b41bd2b6712fe1dd87a98994d4e8323ac8da9c23535eb1f13e96aed8e97b39","family_id":"qf-mod32-topology-interior-membership","seed":202611012090,"semantic_template":"qf-mod32-topology-interior-membership/{\"n\":5,\"point\":4,\"subset\":[2,3],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,2,3],[0,3],[0]]}"},"fields":{"title":"Interior Membership Needs an Open Neighborhood inside the Subset","stem":"Use the complete finite topology=[[], [0], [0, 3], [0, 2, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[2, 3], point=4. Give (point,interior,in_interior?,topology_relative_interior).","answer":{"kind":"expression","value":"(4, [], No, topology_relative_interior)","explanation":"Unite all opens contained in the subset, then test the point. Membership in the subset alone need not give membership in its interior. The exact result, retaining the requested order, is (4, [], No, topology_relative_interior)."},"skills":["Interior Membership Needs an Open Neighborhood inside the Subset"],"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-20261101-r02-closure-and-interior-0091","batch_id":"batch-20261014-advance-20261101-r02-closure-and-interior-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d0f858da3bb3c74f40b76b43fb0141b2b2b17c7345a0d0e18340dc4a397becad","family_id":"qf-mod32-topology-interior-membership","seed":202611012091,"semantic_template":"qf-mod32-topology-interior-membership/{\"n\":5,\"point\":0,\"subset\":[2,3],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,3],[0,3],[0]]}"},"fields":{"title":"Interior Membership Needs an Open Neighborhood inside the Subset","stem":"Use the complete finite topology=[[], [0], [0, 3], [0, 2, 3], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[2, 3], point=0. Give (point,interior,in_interior?,topology_relative_interior).","answer":{"kind":"expression","value":"(0, [], No, topology_relative_interior)","explanation":"Unite all opens contained in the subset, then test the point. Membership in the subset alone need not give membership in its interior. The exact result, retaining the requested order, is (0, [], No, topology_relative_interior)."},"skills":["Interior Membership Needs an Open Neighborhood inside the Subset"],"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-20261101-r02-closure-and-interior-0092","batch_id":"batch-20261014-advance-20261101-r02-closure-and-interior-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"25e91cf57d9e7b3b5161e9531cc9382add84ae9f7453b94fcefd31603d79561f","family_id":"qf-mod32-topology-interior-membership","seed":202611012092,"semantic_template":"qf-mod32-topology-interior-membership/{\"n\":5,\"point\":3,\"subset\":[3,4],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,3,4],[0,3],[0]]}"},"fields":{"title":"Interior Membership Needs an Open Neighborhood inside the Subset","stem":"Use the complete finite topology=[[], [0], [0, 3], [0, 3, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[3, 4], point=3. Give (point,interior,in_interior?,topology_relative_interior).","answer":{"kind":"expression","value":"(3, [], No, topology_relative_interior)","explanation":"Unite all opens contained in the subset, then test the point. Membership in the subset alone need not give membership in its interior. The exact result, retaining the requested order, is (3, [], No, topology_relative_interior)."},"skills":["Interior Membership Needs an Open Neighborhood inside the Subset"],"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-20261101-r02-closure-and-interior-0093","batch_id":"batch-20261014-advance-20261101-r02-closure-and-interior-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"143802e8d7e8ecf9370ee744bb40614926d7432fcf9af69f957b0a40a57ca5ef","family_id":"qf-mod32-topology-interior-membership","seed":202611012093,"semantic_template":"qf-mod32-topology-interior-membership/{\"n\":5,\"point\":4,\"subset\":[3,4],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,3,4],[0,3],[0]]}"},"fields":{"title":"Interior Membership Needs an Open Neighborhood inside the Subset","stem":"Use the complete finite topology=[[], [0], [0, 3], [0, 3, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[3, 4], point=4. Give (point,interior,in_interior?,topology_relative_interior).","answer":{"kind":"expression","value":"(4, [], No, topology_relative_interior)","explanation":"Unite all opens contained in the subset, then test the point. Membership in the subset alone need not give membership in its interior. The exact result, retaining the requested order, is (4, [], No, topology_relative_interior)."},"skills":["Interior Membership Needs an Open Neighborhood inside the Subset"],"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-20261101-r02-closure-and-interior-0094","batch_id":"batch-20261014-advance-20261101-r02-closure-and-interior-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e3b31382a0afb693aadc1e11befbfd71267f6e942e35d3e0735d8d94ce3d2aa9","family_id":"qf-mod32-topology-interior-membership","seed":202611012094,"semantic_template":"qf-mod32-topology-interior-membership/{\"n\":5,\"point\":2,\"subset\":[1,4],\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,4],[0,4],[0]]}"},"fields":{"title":"Interior Membership Needs an Open Neighborhood inside the Subset","stem":"Use the complete finite topology=[[], [0], [0, 4], [0, 1, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[1, 4], point=2. Give (point,interior,in_interior?,topology_relative_interior).","answer":{"kind":"expression","value":"(2, [], No, topology_relative_interior)","explanation":"Unite all opens contained in the subset, then test the point. Membership in the subset alone need not give membership in its interior. The exact result, retaining the requested order, is (2, [], No, topology_relative_interior)."},"skills":["Interior Membership Needs an Open Neighborhood inside the Subset"],"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-20261101-r02-closure-and-interior-0095","batch_id":"batch-20261014-advance-20261101-r02-closure-and-interior-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cbe24ad4dad0267e6ba9e1220fa2344bfefe53b503272d9c42996e80ac8240d0","family_id":"qf-mod32-topology-interior-membership","seed":202611012095,"semantic_template":"qf-mod32-topology-interior-membership/{\"n\":5,\"point\":2,\"subset\":[1,4],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,4],[0,4],[0]]}"},"fields":{"title":"Interior Membership Needs an Open Neighborhood inside the Subset","stem":"Use the complete finite topology=[[], [0], [0, 4], [0, 1, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[1, 4], point=2. Give (point,interior,in_interior?,topology_relative_interior).","answer":{"kind":"expression","value":"(2, [], No, topology_relative_interior)","explanation":"Unite all opens contained in the subset, then test the point. Membership in the subset alone need not give membership in its interior. The exact result, retaining the requested order, is (2, [], No, topology_relative_interior)."},"skills":["Interior Membership Needs an Open Neighborhood inside the Subset"],"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-20261101-r02-closure-and-interior-0096","batch_id":"batch-20261014-advance-20261101-r02-closure-and-interior-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2c10e8b70cb99cc1538d8e120aed91fe3136e04d990a9edcc3e3a3911baa66cb","family_id":"qf-mod32-topology-interior-membership","seed":202611012096,"semantic_template":"qf-mod32-topology-interior-membership/{\"n\":5,\"point\":0,\"subset\":[2,4],\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,2,4],[0,4],[0]]}"},"fields":{"title":"Interior Membership Needs an Open Neighborhood inside the Subset","stem":"Use the complete finite topology=[[], [0], [0, 4], [0, 2, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[2, 4], point=0. Give (point,interior,in_interior?,topology_relative_interior).","answer":{"kind":"expression","value":"(0, [], No, topology_relative_interior)","explanation":"Unite all opens contained in the subset, then test the point. Membership in the subset alone need not give membership in its interior. The exact result, retaining the requested order, is (0, [], No, topology_relative_interior)."},"skills":["Interior Membership Needs an Open Neighborhood inside the Subset"],"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-20261101-r02-closure-and-interior-0097","batch_id":"batch-20261014-advance-20261101-r02-closure-and-interior-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e9b0d3866b59d097c5d24f0de5bf779fc49151f8824b99f4e10f47a066469b84","family_id":"qf-mod32-topology-interior-membership","seed":202611012097,"semantic_template":"qf-mod32-topology-interior-membership/{\"n\":5,\"point\":4,\"subset\":[2,4],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,4],[0,4],[0]]}"},"fields":{"title":"Interior Membership Needs an Open Neighborhood inside the Subset","stem":"Use the complete finite topology=[[], [0], [0, 4], [0, 2, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[2, 4], point=4. Give (point,interior,in_interior?,topology_relative_interior).","answer":{"kind":"expression","value":"(4, [], No, topology_relative_interior)","explanation":"Unite all opens contained in the subset, then test the point. Membership in the subset alone need not give membership in its interior. The exact result, retaining the requested order, is (4, [], No, topology_relative_interior)."},"skills":["Interior Membership Needs an Open Neighborhood inside the Subset"],"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-20261101-r02-closure-and-interior-0098","batch_id":"batch-20261014-advance-20261101-r02-closure-and-interior-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b126ca90cd5d4805a79b72ce59d73efb3548f2308c79834f086f82a4caf3a7db","family_id":"qf-mod32-topology-interior-membership","seed":202611012098,"semantic_template":"qf-mod32-topology-interior-membership/{\"n\":5,\"point\":3,\"subset\":[3,4],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,3,4],[0,4],[0]]}"},"fields":{"title":"Interior Membership Needs an Open Neighborhood inside the Subset","stem":"Use the complete finite topology=[[], [0], [0, 4], [0, 3, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[3, 4], point=3. Give (point,interior,in_interior?,topology_relative_interior).","answer":{"kind":"expression","value":"(3, [], No, topology_relative_interior)","explanation":"Unite all opens contained in the subset, then test the point. Membership in the subset alone need not give membership in its interior. The exact result, retaining the requested order, is (3, [], No, topology_relative_interior)."},"skills":["Interior Membership Needs an Open Neighborhood inside the Subset"],"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-20261101-r02-closure-and-interior-0099","batch_id":"batch-20261014-advance-20261101-r02-closure-and-interior-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"19f8231bb752775a88fbf150128ea7585f97e74164b5c61d33239106c82f42d3","family_id":"qf-mod32-topology-interior-membership","seed":202611012099,"semantic_template":"qf-mod32-topology-interior-membership/{\"n\":5,\"point\":2,\"subset\":[3,4],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,3,4],[0,4],[0]]}"},"fields":{"title":"Interior Membership Needs an Open Neighborhood inside the Subset","stem":"Use the complete finite topology=[[], [0], [0, 4], [0, 3, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[3, 4], point=2. Give (point,interior,in_interior?,topology_relative_interior).","answer":{"kind":"expression","value":"(2, [], No, topology_relative_interior)","explanation":"Unite all opens contained in the subset, then test the point. Membership in the subset alone need not give membership in its interior. The exact result, retaining the requested order, is (2, [], No, topology_relative_interior)."},"skills":["Interior Membership Needs an Open Neighborhood inside the Subset"],"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-20261101-r02-closure-and-interior-0100","batch_id":"batch-20261014-advance-20261101-r02-closure-and-interior-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dc7a6abcbccb0a6006ccff4ce79f21f9120819bfd7789c6d270e886882924b57","family_id":"qf-mod32-topology-interior-membership","seed":202611012100,"semantic_template":"qf-mod32-topology-interior-membership/{\"n\":5,\"point\":4,\"subset\":[0,2],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2],[0,1],[1]]}"},"fields":{"title":"Interior Membership Needs an Open Neighborhood inside the Subset","stem":"Use the complete finite topology=[[], [1], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[0, 2], point=4. Give (point,interior,in_interior?,topology_relative_interior).","answer":{"kind":"expression","value":"(4, [], No, topology_relative_interior)","explanation":"Unite all opens contained in the subset, then test the point. Membership in the subset alone need not give membership in its interior. The exact result, retaining the requested order, is (4, [], No, topology_relative_interior)."},"skills":["Interior Membership Needs an Open Neighborhood inside the Subset"],"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."}}