Hausdorff Separation, Finite Subcovers and Ambient Closedness
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batch-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-foundations · 2026-10-14 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
Field order uses a source-bound presentation supplement; canonical JSONL and mathematical answers are unchanged.
Field-order and source-binding data
1. Ambient Hausdorffness and Compactness Are Both Theorem Hypotheses
Use complete finite labeled carrier=[a0, b0, c0], topology=[[a0, b0, c0], [a0, b0], [a0, c0], [a0], [b0, c0], [b0], [c0], []] in its original supplied open-set order, and compact_subset K=[a0, b0]. Qualify the topology and every Hausdorff distinct-point pair. Finite K is compact; the general theorem still needs both ambient Hausdorffness and compactness. Give (points, compact_subset, pair_separation_rows, Hausdorff, compact, certificate).
View answer and explanation
([a0, b0, c0], [a0, b0], [[a0, b0, [[a0, c0], [b0]]], [a0, c0, [[a0, b0], [c0]]], [b0, c0, [[a0, b0], [c0]]]], Yes, Yes, complete_finite_topology_Hausdorff_and_finite_subset_compact)
Require every distinct-point pair to have disjoint open neighborhoods, retaining first witnesses in supplied topology order. Report Hausdorffness and finite-subset compactness separately; do not drop the Hausdorff hypothesis in the general theorem. The exact result, retaining the requested order, is ([a0, b0, c0], [a0, b0], [[a0, b0, [[a0, c0], [b0]]], [a0, c0, [[a0, b0], [c0]]], [b0, c0, [[a0, b0], [c0]]]], Yes, Yes, complete_finite_topology_Hausdorff_and_finite_subset_compact).
Reviewed step-by-step derivation
A compact subset of a Hausdorff space is closed · qf-mod43-compactclosed-hypotheses-qualification · candidate
2. Ambient Hausdorffness and Compactness Are Both Theorem Hypotheses
Use complete finite labeled carrier=[a1, b1, c1], topology=[[a1, b1, c1], [a1, b1], [a1, c1], [a1], [b1, c1], [b1], [c1], []] in its original supplied open-set order, and compact_subset K=[a1, b1]. Qualify the topology and every Hausdorff distinct-point pair. Finite K is compact; the general theorem still needs both ambient Hausdorffness and compactness. Give (points, compact_subset, pair_separation_rows, Hausdorff, compact, certificate).
View answer and explanation
([a1, b1, c1], [a1, b1], [[a1, b1, [[a1, c1], [b1]]], [a1, c1, [[a1, b1], [c1]]], [b1, c1, [[a1, b1], [c1]]]], Yes, Yes, complete_finite_topology_Hausdorff_and_finite_subset_compact)
Require every distinct-point pair to have disjoint open neighborhoods, retaining first witnesses in supplied topology order. Report Hausdorffness and finite-subset compactness separately; do not drop the Hausdorff hypothesis in the general theorem. The exact result, retaining the requested order, is ([a1, b1, c1], [a1, b1], [[a1, b1, [[a1, c1], [b1]]], [a1, c1, [[a1, b1], [c1]]], [b1, c1, [[a1, b1], [c1]]]], Yes, Yes, complete_finite_topology_Hausdorff_and_finite_subset_compact).
Reviewed step-by-step derivation
A compact subset of a Hausdorff space is closed · qf-mod43-compactclosed-hypotheses-qualification · candidate
3. Ambient Hausdorffness and Compactness Are Both Theorem Hypotheses
Use complete finite labeled carrier=[a2, b2, c2], topology=[[a2, b2, c2], [a2, b2], [a2, c2], [a2], [b2, c2], [b2], [c2], []] in its original supplied open-set order, and compact_subset K=[a2, b2]. Qualify the topology and every Hausdorff distinct-point pair. Finite K is compact; the general theorem still needs both ambient Hausdorffness and compactness. Give (points, compact_subset, pair_separation_rows, Hausdorff, compact, certificate).
View answer and explanation
([a2, b2, c2], [a2, b2], [[a2, b2, [[a2, c2], [b2]]], [a2, c2, [[a2, b2], [c2]]], [b2, c2, [[a2, b2], [c2]]]], Yes, Yes, complete_finite_topology_Hausdorff_and_finite_subset_compact)
Require every distinct-point pair to have disjoint open neighborhoods, retaining first witnesses in supplied topology order. Report Hausdorffness and finite-subset compactness separately; do not drop the Hausdorff hypothesis in the general theorem. The exact result, retaining the requested order, is ([a2, b2, c2], [a2, b2], [[a2, b2, [[a2, c2], [b2]]], [a2, c2, [[a2, b2], [c2]]], [b2, c2, [[a2, b2], [c2]]]], Yes, Yes, complete_finite_topology_Hausdorff_and_finite_subset_compact).
Reviewed step-by-step derivation
A compact subset of a Hausdorff space is closed · qf-mod43-compactclosed-hypotheses-qualification · candidate
4. Ambient Hausdorffness and Compactness Are Both Theorem Hypotheses
Use complete finite labeled carrier=[a3, b3, c3], topology=[[a3, b3, c3], [a3, b3], [a3, c3], [a3], [b3, c3], [b3], [c3], []] in its original supplied open-set order, and compact_subset K=[a3, b3]. Qualify the topology and every Hausdorff distinct-point pair. Finite K is compact; the general theorem still needs both ambient Hausdorffness and compactness. Give (points, compact_subset, pair_separation_rows, Hausdorff, compact, certificate).
View answer and explanation
([a3, b3, c3], [a3, b3], [[a3, b3, [[a3, c3], [b3]]], [a3, c3, [[a3, b3], [c3]]], [b3, c3, [[a3, b3], [c3]]]], Yes, Yes, complete_finite_topology_Hausdorff_and_finite_subset_compact)
Require every distinct-point pair to have disjoint open neighborhoods, retaining first witnesses in supplied topology order. Report Hausdorffness and finite-subset compactness separately; do not drop the Hausdorff hypothesis in the general theorem. The exact result, retaining the requested order, is ([a3, b3, c3], [a3, b3], [[a3, b3, [[a3, c3], [b3]]], [a3, c3, [[a3, b3], [c3]]], [b3, c3, [[a3, b3], [c3]]]], Yes, Yes, complete_finite_topology_Hausdorff_and_finite_subset_compact).
Reviewed step-by-step derivation
A compact subset of a Hausdorff space is closed · qf-mod43-compactclosed-hypotheses-qualification · candidate
5. Ambient Hausdorffness and Compactness Are Both Theorem Hypotheses
Use complete finite labeled carrier=[a4, b4, c4], topology=[[a4, b4, c4], [a4, b4], [a4, c4], [a4], [b4, c4], [b4], [c4], []] in its original supplied open-set order, and compact_subset K=[a4, b4]. Qualify the topology and every Hausdorff distinct-point pair. Finite K is compact; the general theorem still needs both ambient Hausdorffness and compactness. Give (points, compact_subset, pair_separation_rows, Hausdorff, compact, certificate).
View answer and explanation
([a4, b4, c4], [a4, b4], [[a4, b4, [[a4, c4], [b4]]], [a4, c4, [[a4, b4], [c4]]], [b4, c4, [[a4, b4], [c4]]]], Yes, Yes, complete_finite_topology_Hausdorff_and_finite_subset_compact)
Require every distinct-point pair to have disjoint open neighborhoods, retaining first witnesses in supplied topology order. Report Hausdorffness and finite-subset compactness separately; do not drop the Hausdorff hypothesis in the general theorem. The exact result, retaining the requested order, is ([a4, b4, c4], [a4, b4], [[a4, b4, [[a4, c4], [b4]]], [a4, c4, [[a4, b4], [c4]]], [b4, c4, [[a4, b4], [c4]]]], Yes, Yes, complete_finite_topology_Hausdorff_and_finite_subset_compact).
Reviewed step-by-step derivation
A compact subset of a Hausdorff space is closed · qf-mod43-compactclosed-hypotheses-qualification · candidate
6. Ambient Hausdorffness and Compactness Are Both Theorem Hypotheses
Use complete finite labeled carrier=[a5, b5, c5], topology=[[a5, b5, c5], [a5, b5], [a5, c5], [a5], [b5, c5], [b5], [c5], []] in its original supplied open-set order, and compact_subset K=[a5, b5]. Qualify the topology and every Hausdorff distinct-point pair. Finite K is compact; the general theorem still needs both ambient Hausdorffness and compactness. Give (points, compact_subset, pair_separation_rows, Hausdorff, compact, certificate).
View answer and explanation
([a5, b5, c5], [a5, b5], [[a5, b5, [[a5, c5], [b5]]], [a5, c5, [[a5, b5], [c5]]], [b5, c5, [[a5, b5], [c5]]]], Yes, Yes, complete_finite_topology_Hausdorff_and_finite_subset_compact)
Require every distinct-point pair to have disjoint open neighborhoods, retaining first witnesses in supplied topology order. Report Hausdorffness and finite-subset compactness separately; do not drop the Hausdorff hypothesis in the general theorem. The exact result, retaining the requested order, is ([a5, b5, c5], [a5, b5], [[a5, b5, [[a5, c5], [b5]]], [a5, c5, [[a5, b5], [c5]]], [b5, c5, [[a5, b5], [c5]]]], Yes, Yes, complete_finite_topology_Hausdorff_and_finite_subset_compact).
Reviewed step-by-step derivation
A compact subset of a Hausdorff space is closed · qf-mod43-compactclosed-hypotheses-qualification · candidate
7. Ambient Hausdorffness and Compactness Are Both Theorem Hypotheses
Use complete finite labeled carrier=[a6, b6, c6], topology=[[a6, b6, c6], [a6, b6], [a6, c6], [a6], [b6, c6], [b6], [c6], []] in its original supplied open-set order, and compact_subset K=[a6, b6]. Qualify the topology and every Hausdorff distinct-point pair. Finite K is compact; the general theorem still needs both ambient Hausdorffness and compactness. Give (points, compact_subset, pair_separation_rows, Hausdorff, compact, certificate).
View answer and explanation
([a6, b6, c6], [a6, b6], [[a6, b6, [[a6, c6], [b6]]], [a6, c6, [[a6, b6], [c6]]], [b6, c6, [[a6, b6], [c6]]]], Yes, Yes, complete_finite_topology_Hausdorff_and_finite_subset_compact)
Require every distinct-point pair to have disjoint open neighborhoods, retaining first witnesses in supplied topology order. Report Hausdorffness and finite-subset compactness separately; do not drop the Hausdorff hypothesis in the general theorem. The exact result, retaining the requested order, is ([a6, b6, c6], [a6, b6], [[a6, b6, [[a6, c6], [b6]]], [a6, c6, [[a6, b6], [c6]]], [b6, c6, [[a6, b6], [c6]]]], Yes, Yes, complete_finite_topology_Hausdorff_and_finite_subset_compact).
Reviewed step-by-step derivation
A compact subset of a Hausdorff space is closed · qf-mod43-compactclosed-hypotheses-qualification · candidate
8. Ambient Hausdorffness and Compactness Are Both Theorem Hypotheses
Use complete finite labeled carrier=[a7, b7, c7], topology=[[a7, b7, c7], [a7, b7], [a7, c7], [a7], [b7, c7], [b7], [c7], []] in its original supplied open-set order, and compact_subset K=[a7, b7]. Qualify the topology and every Hausdorff distinct-point pair. Finite K is compact; the general theorem still needs both ambient Hausdorffness and compactness. Give (points, compact_subset, pair_separation_rows, Hausdorff, compact, certificate).
View answer and explanation
([a7, b7, c7], [a7, b7], [[a7, b7, [[a7, c7], [b7]]], [a7, c7, [[a7, b7], [c7]]], [b7, c7, [[a7, b7], [c7]]]], Yes, Yes, complete_finite_topology_Hausdorff_and_finite_subset_compact)
Require every distinct-point pair to have disjoint open neighborhoods, retaining first witnesses in supplied topology order. Report Hausdorffness and finite-subset compactness separately; do not drop the Hausdorff hypothesis in the general theorem. The exact result, retaining the requested order, is ([a7, b7, c7], [a7, b7], [[a7, b7, [[a7, c7], [b7]]], [a7, c7, [[a7, b7], [c7]]], [b7, c7, [[a7, b7], [c7]]]], Yes, Yes, complete_finite_topology_Hausdorff_and_finite_subset_compact).
Reviewed step-by-step derivation
A compact subset of a Hausdorff space is closed · qf-mod43-compactclosed-hypotheses-qualification · candidate
9. Ambient Hausdorffness and Compactness Are Both Theorem Hypotheses
Use complete finite labeled carrier=[a8, b8, c8], topology=[[a8, b8, c8], [a8, b8], [a8, c8], [a8], [b8, c8], [b8], [c8], []] in its original supplied open-set order, and compact_subset K=[a8, b8]. Qualify the topology and every Hausdorff distinct-point pair. Finite K is compact; the general theorem still needs both ambient Hausdorffness and compactness. Give (points, compact_subset, pair_separation_rows, Hausdorff, compact, certificate).
View answer and explanation
([a8, b8, c8], [a8, b8], [[a8, b8, [[a8, c8], [b8]]], [a8, c8, [[a8, b8], [c8]]], [b8, c8, [[a8, b8], [c8]]]], Yes, Yes, complete_finite_topology_Hausdorff_and_finite_subset_compact)
Require every distinct-point pair to have disjoint open neighborhoods, retaining first witnesses in supplied topology order. Report Hausdorffness and finite-subset compactness separately; do not drop the Hausdorff hypothesis in the general theorem. The exact result, retaining the requested order, is ([a8, b8, c8], [a8, b8], [[a8, b8, [[a8, c8], [b8]]], [a8, c8, [[a8, b8], [c8]]], [b8, c8, [[a8, b8], [c8]]]], Yes, Yes, complete_finite_topology_Hausdorff_and_finite_subset_compact).
Reviewed step-by-step derivation
A compact subset of a Hausdorff space is closed · qf-mod43-compactclosed-hypotheses-qualification · candidate
10. Ambient Hausdorffness and Compactness Are Both Theorem Hypotheses
Use complete finite labeled carrier=[a9, b9, c9], topology=[[a9, b9, c9], [a9, b9], [a9, c9], [a9], [b9, c9], [b9], [c9], []] in its original supplied open-set order, and compact_subset K=[a9, b9]. Qualify the topology and every Hausdorff distinct-point pair. Finite K is compact; the general theorem still needs both ambient Hausdorffness and compactness. Give (points, compact_subset, pair_separation_rows, Hausdorff, compact, certificate).
View answer and explanation
([a9, b9, c9], [a9, b9], [[a9, b9, [[a9, c9], [b9]]], [a9, c9, [[a9, b9], [c9]]], [b9, c9, [[a9, b9], [c9]]]], Yes, Yes, complete_finite_topology_Hausdorff_and_finite_subset_compact)
Require every distinct-point pair to have disjoint open neighborhoods, retaining first witnesses in supplied topology order. Report Hausdorffness and finite-subset compactness separately; do not drop the Hausdorff hypothesis in the general theorem. The exact result, retaining the requested order, is ([a9, b9, c9], [a9, b9], [[a9, b9, [[a9, c9], [b9]]], [a9, c9, [[a9, b9], [c9]]], [b9, c9, [[a9, b9], [c9]]]], Yes, Yes, complete_finite_topology_Hausdorff_and_finite_subset_compact).
Reviewed step-by-step derivation
A compact subset of a Hausdorff space is closed · qf-mod43-compactclosed-hypotheses-qualification · candidate
11. Ambient Hausdorffness and Compactness Are Both Theorem Hypotheses
Use complete finite labeled carrier=[a10, b10, c10], topology=[[a10, b10, c10], [a10, b10], [a10, c10], [a10], [b10, c10], [b10], [c10], []] in its original supplied open-set order, and compact_subset K=[a10, b10]. Qualify the topology and every Hausdorff distinct-point pair. Finite K is compact; the general theorem still needs both ambient Hausdorffness and compactness. Give (points, compact_subset, pair_separation_rows, Hausdorff, compact, certificate).
View answer and explanation
([a10, b10, c10], [a10, b10], [[a10, b10, [[a10, c10], [b10]]], [a10, c10, [[a10, b10], [c10]]], [b10, c10, [[a10, b10], [c10]]]], Yes, Yes, complete_finite_topology_Hausdorff_and_finite_subset_compact)
Require every distinct-point pair to have disjoint open neighborhoods, retaining first witnesses in supplied topology order. Report Hausdorffness and finite-subset compactness separately; do not drop the Hausdorff hypothesis in the general theorem. The exact result, retaining the requested order, is ([a10, b10, c10], [a10, b10], [[a10, b10, [[a10, c10], [b10]]], [a10, c10, [[a10, b10], [c10]]], [b10, c10, [[a10, b10], [c10]]]], Yes, Yes, complete_finite_topology_Hausdorff_and_finite_subset_compact).
Reviewed step-by-step derivation
A compact subset of a Hausdorff space is closed · qf-mod43-compactclosed-hypotheses-qualification · candidate
12. Ambient Hausdorffness and Compactness Are Both Theorem Hypotheses
Use complete finite labeled carrier=[a11, b11, c11], topology=[[a11, b11, c11], [a11, b11], [a11, c11], [a11], [b11, c11], [b11], [c11], []] in its original supplied open-set order, and compact_subset K=[a11, b11]. Qualify the topology and every Hausdorff distinct-point pair. Finite K is compact; the general theorem still needs both ambient Hausdorffness and compactness. Give (points, compact_subset, pair_separation_rows, Hausdorff, compact, certificate).
View answer and explanation
([a11, b11, c11], [a11, b11], [[a11, b11, [[a11, c11], [b11]]], [a11, c11, [[a11, b11], [c11]]], [b11, c11, [[a11, b11], [c11]]]], Yes, Yes, complete_finite_topology_Hausdorff_and_finite_subset_compact)
Require every distinct-point pair to have disjoint open neighborhoods, retaining first witnesses in supplied topology order. Report Hausdorffness and finite-subset compactness separately; do not drop the Hausdorff hypothesis in the general theorem. The exact result, retaining the requested order, is ([a11, b11, c11], [a11, b11], [[a11, b11, [[a11, c11], [b11]]], [a11, c11, [[a11, b11], [c11]]], [b11, c11, [[a11, b11], [c11]]]], Yes, Yes, complete_finite_topology_Hausdorff_and_finite_subset_compact).
Reviewed step-by-step derivation
A compact subset of a Hausdorff space is closed · qf-mod43-compactclosed-hypotheses-qualification · candidate
13. Ambient Hausdorffness and Compactness Are Both Theorem Hypotheses
Use complete finite labeled carrier=[a12, b12, c12], topology=[[a12, b12, c12], [a12, b12], [a12, c12], [a12], [b12, c12], [b12], [c12], []] in its original supplied open-set order, and compact_subset K=[a12, b12]. Qualify the topology and every Hausdorff distinct-point pair. Finite K is compact; the general theorem still needs both ambient Hausdorffness and compactness. Give (points, compact_subset, pair_separation_rows, Hausdorff, compact, certificate).
View answer and explanation
([a12, b12, c12], [a12, b12], [[a12, b12, [[a12, c12], [b12]]], [a12, c12, [[a12, b12], [c12]]], [b12, c12, [[a12, b12], [c12]]]], Yes, Yes, complete_finite_topology_Hausdorff_and_finite_subset_compact)
Require every distinct-point pair to have disjoint open neighborhoods, retaining first witnesses in supplied topology order. Report Hausdorffness and finite-subset compactness separately; do not drop the Hausdorff hypothesis in the general theorem. The exact result, retaining the requested order, is ([a12, b12, c12], [a12, b12], [[a12, b12, [[a12, c12], [b12]]], [a12, c12, [[a12, b12], [c12]]], [b12, c12, [[a12, b12], [c12]]]], Yes, Yes, complete_finite_topology_Hausdorff_and_finite_subset_compact).
Reviewed step-by-step derivation
A compact subset of a Hausdorff space is closed · qf-mod43-compactclosed-hypotheses-qualification · candidate
14. Ambient Hausdorffness and Compactness Are Both Theorem Hypotheses
Use complete finite labeled carrier=[a13, b13, c13], topology=[[a13, b13, c13], [a13, b13], [a13, c13], [a13], [b13, c13], [b13], [c13], []] in its original supplied open-set order, and compact_subset K=[a13, b13]. Qualify the topology and every Hausdorff distinct-point pair. Finite K is compact; the general theorem still needs both ambient Hausdorffness and compactness. Give (points, compact_subset, pair_separation_rows, Hausdorff, compact, certificate).
View answer and explanation
([a13, b13, c13], [a13, b13], [[a13, b13, [[a13, c13], [b13]]], [a13, c13, [[a13, b13], [c13]]], [b13, c13, [[a13, b13], [c13]]]], Yes, Yes, complete_finite_topology_Hausdorff_and_finite_subset_compact)
Require every distinct-point pair to have disjoint open neighborhoods, retaining first witnesses in supplied topology order. Report Hausdorffness and finite-subset compactness separately; do not drop the Hausdorff hypothesis in the general theorem. The exact result, retaining the requested order, is ([a13, b13, c13], [a13, b13], [[a13, b13, [[a13, c13], [b13]]], [a13, c13, [[a13, b13], [c13]]], [b13, c13, [[a13, b13], [c13]]]], Yes, Yes, complete_finite_topology_Hausdorff_and_finite_subset_compact).
Reviewed step-by-step derivation
A compact subset of a Hausdorff space is closed · qf-mod43-compactclosed-hypotheses-qualification · candidate
15. Ambient Hausdorffness and Compactness Are Both Theorem Hypotheses
Use complete finite labeled carrier=[a14, b14, c14], topology=[[a14, b14, c14], [a14, b14], [a14, c14], [a14], [b14, c14], [b14], [c14], []] in its original supplied open-set order, and compact_subset K=[a14, b14]. Qualify the topology and every Hausdorff distinct-point pair. Finite K is compact; the general theorem still needs both ambient Hausdorffness and compactness. Give (points, compact_subset, pair_separation_rows, Hausdorff, compact, certificate).
View answer and explanation
([a14, b14, c14], [a14, b14], [[a14, b14, [[a14, c14], [b14]]], [a14, c14, [[a14, b14], [c14]]], [b14, c14, [[a14, b14], [c14]]]], Yes, Yes, complete_finite_topology_Hausdorff_and_finite_subset_compact)
Require every distinct-point pair to have disjoint open neighborhoods, retaining first witnesses in supplied topology order. Report Hausdorffness and finite-subset compactness separately; do not drop the Hausdorff hypothesis in the general theorem. The exact result, retaining the requested order, is ([a14, b14, c14], [a14, b14], [[a14, b14, [[a14, c14], [b14]]], [a14, c14, [[a14, b14], [c14]]], [b14, c14, [[a14, b14], [c14]]]], Yes, Yes, complete_finite_topology_Hausdorff_and_finite_subset_compact).
Reviewed step-by-step derivation
A compact subset of a Hausdorff space is closed · qf-mod43-compactclosed-hypotheses-qualification · candidate
16. Ambient Hausdorffness and Compactness Are Both Theorem Hypotheses
Use complete finite labeled carrier=[a15, b15, c15], topology=[[a15, b15, c15], [a15, b15], [a15, c15], [a15], [b15, c15], [b15], [c15], []] in its original supplied open-set order, and compact_subset K=[a15, b15]. Qualify the topology and every Hausdorff distinct-point pair. Finite K is compact; the general theorem still needs both ambient Hausdorffness and compactness. Give (points, compact_subset, pair_separation_rows, Hausdorff, compact, certificate).
View answer and explanation
([a15, b15, c15], [a15, b15], [[a15, b15, [[a15, c15], [b15]]], [a15, c15, [[a15, b15], [c15]]], [b15, c15, [[a15, b15], [c15]]]], Yes, Yes, complete_finite_topology_Hausdorff_and_finite_subset_compact)
Require every distinct-point pair to have disjoint open neighborhoods, retaining first witnesses in supplied topology order. Report Hausdorffness and finite-subset compactness separately; do not drop the Hausdorff hypothesis in the general theorem. The exact result, retaining the requested order, is ([a15, b15, c15], [a15, b15], [[a15, b15, [[a15, c15], [b15]]], [a15, c15, [[a15, b15], [c15]]], [b15, c15, [[a15, b15], [c15]]]], Yes, Yes, complete_finite_topology_Hausdorff_and_finite_subset_compact).
Reviewed step-by-step derivation
A compact subset of a Hausdorff space is closed · qf-mod43-compactclosed-hypotheses-qualification · candidate
17. Ambient Hausdorffness and Compactness Are Both Theorem Hypotheses
Use complete finite labeled carrier=[a16, b16, c16], topology=[[a16, b16, c16], [a16, b16], [a16, c16], [a16], [b16, c16], [b16], [c16], []] in its original supplied open-set order, and compact_subset K=[a16, b16]. Qualify the topology and every Hausdorff distinct-point pair. Finite K is compact; the general theorem still needs both ambient Hausdorffness and compactness. Give (points, compact_subset, pair_separation_rows, Hausdorff, compact, certificate).
View answer and explanation
([a16, b16, c16], [a16, b16], [[a16, b16, [[a16, c16], [b16]]], [a16, c16, [[a16, b16], [c16]]], [b16, c16, [[a16, b16], [c16]]]], Yes, Yes, complete_finite_topology_Hausdorff_and_finite_subset_compact)
Require every distinct-point pair to have disjoint open neighborhoods, retaining first witnesses in supplied topology order. Report Hausdorffness and finite-subset compactness separately; do not drop the Hausdorff hypothesis in the general theorem. The exact result, retaining the requested order, is ([a16, b16, c16], [a16, b16], [[a16, b16, [[a16, c16], [b16]]], [a16, c16, [[a16, b16], [c16]]], [b16, c16, [[a16, b16], [c16]]]], Yes, Yes, complete_finite_topology_Hausdorff_and_finite_subset_compact).
Reviewed step-by-step derivation
A compact subset of a Hausdorff space is closed · qf-mod43-compactclosed-hypotheses-qualification · candidate
18. Ambient Hausdorffness and Compactness Are Both Theorem Hypotheses
Use complete finite labeled carrier=[a17, b17, c17], topology=[[a17, b17, c17], [a17, b17], [a17, c17], [a17], [b17, c17], [b17], [c17], []] in its original supplied open-set order, and compact_subset K=[a17, b17]. Qualify the topology and every Hausdorff distinct-point pair. Finite K is compact; the general theorem still needs both ambient Hausdorffness and compactness. Give (points, compact_subset, pair_separation_rows, Hausdorff, compact, certificate).
View answer and explanation
([a17, b17, c17], [a17, b17], [[a17, b17, [[a17, c17], [b17]]], [a17, c17, [[a17, b17], [c17]]], [b17, c17, [[a17, b17], [c17]]]], Yes, Yes, complete_finite_topology_Hausdorff_and_finite_subset_compact)
Require every distinct-point pair to have disjoint open neighborhoods, retaining first witnesses in supplied topology order. Report Hausdorffness and finite-subset compactness separately; do not drop the Hausdorff hypothesis in the general theorem. The exact result, retaining the requested order, is ([a17, b17, c17], [a17, b17], [[a17, b17, [[a17, c17], [b17]]], [a17, c17, [[a17, b17], [c17]]], [b17, c17, [[a17, b17], [c17]]]], Yes, Yes, complete_finite_topology_Hausdorff_and_finite_subset_compact).
Reviewed step-by-step derivation
A compact subset of a Hausdorff space is closed · qf-mod43-compactclosed-hypotheses-qualification · candidate
19. Ambient Hausdorffness and Compactness Are Both Theorem Hypotheses
Use complete finite labeled carrier=[a18, b18, c18], topology=[[a18, b18, c18], [a18, b18], [a18, c18], [a18], [b18, c18], [b18], [c18], []] in its original supplied open-set order, and compact_subset K=[a18, b18]. Qualify the topology and every Hausdorff distinct-point pair. Finite K is compact; the general theorem still needs both ambient Hausdorffness and compactness. Give (points, compact_subset, pair_separation_rows, Hausdorff, compact, certificate).
View answer and explanation
([a18, b18, c18], [a18, b18], [[a18, b18, [[a18, c18], [b18]]], [a18, c18, [[a18, b18], [c18]]], [b18, c18, [[a18, b18], [c18]]]], Yes, Yes, complete_finite_topology_Hausdorff_and_finite_subset_compact)
Require every distinct-point pair to have disjoint open neighborhoods, retaining first witnesses in supplied topology order. Report Hausdorffness and finite-subset compactness separately; do not drop the Hausdorff hypothesis in the general theorem. The exact result, retaining the requested order, is ([a18, b18, c18], [a18, b18], [[a18, b18, [[a18, c18], [b18]]], [a18, c18, [[a18, b18], [c18]]], [b18, c18, [[a18, b18], [c18]]]], Yes, Yes, complete_finite_topology_Hausdorff_and_finite_subset_compact).
Reviewed step-by-step derivation
A compact subset of a Hausdorff space is closed · qf-mod43-compactclosed-hypotheses-qualification · candidate
20. Ambient Hausdorffness and Compactness Are Both Theorem Hypotheses
Use complete finite labeled carrier=[a19, b19, c19], topology=[[a19, b19, c19], [a19, b19], [a19, c19], [a19], [b19, c19], [b19], [c19], []] in its original supplied open-set order, and compact_subset K=[a19, b19]. Qualify the topology and every Hausdorff distinct-point pair. Finite K is compact; the general theorem still needs both ambient Hausdorffness and compactness. Give (points, compact_subset, pair_separation_rows, Hausdorff, compact, certificate).
View answer and explanation
([a19, b19, c19], [a19, b19], [[a19, b19, [[a19, c19], [b19]]], [a19, c19, [[a19, b19], [c19]]], [b19, c19, [[a19, b19], [c19]]]], Yes, Yes, complete_finite_topology_Hausdorff_and_finite_subset_compact)
Require every distinct-point pair to have disjoint open neighborhoods, retaining first witnesses in supplied topology order. Report Hausdorffness and finite-subset compactness separately; do not drop the Hausdorff hypothesis in the general theorem. The exact result, retaining the requested order, is ([a19, b19, c19], [a19, b19], [[a19, b19, [[a19, c19], [b19]]], [a19, c19, [[a19, b19], [c19]]], [b19, c19, [[a19, b19], [c19]]]], Yes, Yes, complete_finite_topology_Hausdorff_and_finite_subset_compact).
Reviewed step-by-step derivation
A compact subset of a Hausdorff space is closed · qf-mod43-compactclosed-hypotheses-qualification · candidate
21. Ambient Hausdorffness and Compactness Are Both Theorem Hypotheses
Use complete finite labeled carrier=[a20, b20, c20], topology=[[a20, b20, c20], [a20, b20], [a20, c20], [a20], [b20, c20], [b20], [c20], []] in its original supplied open-set order, and compact_subset K=[a20, b20]. Qualify the topology and every Hausdorff distinct-point pair. Finite K is compact; the general theorem still needs both ambient Hausdorffness and compactness. Give (points, compact_subset, pair_separation_rows, Hausdorff, compact, certificate).
View answer and explanation
([a20, b20, c20], [a20, b20], [[a20, b20, [[a20, c20], [b20]]], [a20, c20, [[a20, b20], [c20]]], [b20, c20, [[a20, b20], [c20]]]], Yes, Yes, complete_finite_topology_Hausdorff_and_finite_subset_compact)
Require every distinct-point pair to have disjoint open neighborhoods, retaining first witnesses in supplied topology order. Report Hausdorffness and finite-subset compactness separately; do not drop the Hausdorff hypothesis in the general theorem. The exact result, retaining the requested order, is ([a20, b20, c20], [a20, b20], [[a20, b20, [[a20, c20], [b20]]], [a20, c20, [[a20, b20], [c20]]], [b20, c20, [[a20, b20], [c20]]]], Yes, Yes, complete_finite_topology_Hausdorff_and_finite_subset_compact).
Reviewed step-by-step derivation
A compact subset of a Hausdorff space is closed · qf-mod43-compactclosed-hypotheses-qualification · candidate
22. Ambient Hausdorffness and Compactness Are Both Theorem Hypotheses
Use complete finite labeled carrier=[a21, b21, c21], topology=[[a21, b21, c21], [a21, b21], [a21, c21], [a21], [b21, c21], [b21], [c21], []] in its original supplied open-set order, and compact_subset K=[a21, b21]. Qualify the topology and every Hausdorff distinct-point pair. Finite K is compact; the general theorem still needs both ambient Hausdorffness and compactness. Give (points, compact_subset, pair_separation_rows, Hausdorff, compact, certificate).
View answer and explanation
([a21, b21, c21], [a21, b21], [[a21, b21, [[a21, c21], [b21]]], [a21, c21, [[a21, b21], [c21]]], [b21, c21, [[a21, b21], [c21]]]], Yes, Yes, complete_finite_topology_Hausdorff_and_finite_subset_compact)
Require every distinct-point pair to have disjoint open neighborhoods, retaining first witnesses in supplied topology order. Report Hausdorffness and finite-subset compactness separately; do not drop the Hausdorff hypothesis in the general theorem. The exact result, retaining the requested order, is ([a21, b21, c21], [a21, b21], [[a21, b21, [[a21, c21], [b21]]], [a21, c21, [[a21, b21], [c21]]], [b21, c21, [[a21, b21], [c21]]]], Yes, Yes, complete_finite_topology_Hausdorff_and_finite_subset_compact).
Reviewed step-by-step derivation
A compact subset of a Hausdorff space is closed · qf-mod43-compactclosed-hypotheses-qualification · candidate
23. Ambient Hausdorffness and Compactness Are Both Theorem Hypotheses
Use complete finite labeled carrier=[a22, b22, c22], topology=[[a22, b22, c22], [a22, b22], [a22, c22], [a22], [b22, c22], [b22], [c22], []] in its original supplied open-set order, and compact_subset K=[a22, b22]. Qualify the topology and every Hausdorff distinct-point pair. Finite K is compact; the general theorem still needs both ambient Hausdorffness and compactness. Give (points, compact_subset, pair_separation_rows, Hausdorff, compact, certificate).
View answer and explanation
([a22, b22, c22], [a22, b22], [[a22, b22, [[a22, c22], [b22]]], [a22, c22, [[a22, b22], [c22]]], [b22, c22, [[a22, b22], [c22]]]], Yes, Yes, complete_finite_topology_Hausdorff_and_finite_subset_compact)
Require every distinct-point pair to have disjoint open neighborhoods, retaining first witnesses in supplied topology order. Report Hausdorffness and finite-subset compactness separately; do not drop the Hausdorff hypothesis in the general theorem. The exact result, retaining the requested order, is ([a22, b22, c22], [a22, b22], [[a22, b22, [[a22, c22], [b22]]], [a22, c22, [[a22, b22], [c22]]], [b22, c22, [[a22, b22], [c22]]]], Yes, Yes, complete_finite_topology_Hausdorff_and_finite_subset_compact).
Reviewed step-by-step derivation
A compact subset of a Hausdorff space is closed · qf-mod43-compactclosed-hypotheses-qualification · candidate
24. Ambient Hausdorffness and Compactness Are Both Theorem Hypotheses
Use complete finite labeled carrier=[a23, b23, c23], topology=[[a23, b23, c23], [a23, b23], [a23, c23], [a23], [b23, c23], [b23], [c23], []] in its original supplied open-set order, and compact_subset K=[a23, b23]. Qualify the topology and every Hausdorff distinct-point pair. Finite K is compact; the general theorem still needs both ambient Hausdorffness and compactness. Give (points, compact_subset, pair_separation_rows, Hausdorff, compact, certificate).
View answer and explanation
([a23, b23, c23], [a23, b23], [[a23, b23, [[a23, c23], [b23]]], [a23, c23, [[a23, b23], [c23]]], [b23, c23, [[a23, b23], [c23]]]], Yes, Yes, complete_finite_topology_Hausdorff_and_finite_subset_compact)
Require every distinct-point pair to have disjoint open neighborhoods, retaining first witnesses in supplied topology order. Report Hausdorffness and finite-subset compactness separately; do not drop the Hausdorff hypothesis in the general theorem. The exact result, retaining the requested order, is ([a23, b23, c23], [a23, b23], [[a23, b23, [[a23, c23], [b23]]], [a23, c23, [[a23, b23], [c23]]], [b23, c23, [[a23, b23], [c23]]]], Yes, Yes, complete_finite_topology_Hausdorff_and_finite_subset_compact).
Reviewed step-by-step derivation
A compact subset of a Hausdorff space is closed · qf-mod43-compactclosed-hypotheses-qualification · candidate
25. Ambient Hausdorffness and Compactness Are Both Theorem Hypotheses
Use complete finite labeled carrier=[a24, b24, c24], topology=[[a24, b24, c24], [a24, b24], [a24, c24], [a24], [b24, c24], [b24], [c24], []] in its original supplied open-set order, and compact_subset K=[a24, b24]. Qualify the topology and every Hausdorff distinct-point pair. Finite K is compact; the general theorem still needs both ambient Hausdorffness and compactness. Give (points, compact_subset, pair_separation_rows, Hausdorff, compact, certificate).
View answer and explanation
([a24, b24, c24], [a24, b24], [[a24, b24, [[a24, c24], [b24]]], [a24, c24, [[a24, b24], [c24]]], [b24, c24, [[a24, b24], [c24]]]], Yes, Yes, complete_finite_topology_Hausdorff_and_finite_subset_compact)
Require every distinct-point pair to have disjoint open neighborhoods, retaining first witnesses in supplied topology order. Report Hausdorffness and finite-subset compactness separately; do not drop the Hausdorff hypothesis in the general theorem. The exact result, retaining the requested order, is ([a24, b24, c24], [a24, b24], [[a24, b24, [[a24, c24], [b24]]], [a24, c24, [[a24, b24], [c24]]], [b24, c24, [[a24, b24], [c24]]]], Yes, Yes, complete_finite_topology_Hausdorff_and_finite_subset_compact).
Reviewed step-by-step derivation
A compact subset of a Hausdorff space is closed · qf-mod43-compactclosed-hypotheses-qualification · candidate
26. Closedness Uses the Open Complement in the Same Ambient Space
Use complete finite labeled carrier=[a0, b0, c0], topology=[[a0, b0, c0], [a0, b0], [a0, c0], [a0], [b0, c0], [b0], [c0], []] in its original supplied open-set order, and compact_subset K=[a0, b0]. Qualify the topology and every Hausdorff distinct-point pair. Finite K is compact; the general theorem still needs both ambient Hausdorffness and compactness. Give (compact_subset, complement, compact, closed, certificate).
View answer and explanation
([a0, b0], [c0], Yes, Yes, complement_is_open_so_compact_subset_is_closed)
Require the positive Hausdorff instance and verify that X minus K is open. Empty K is also closed. Finite model verification does not prove the general theorem or its converse closed-implies-compact. The exact result, retaining the requested order, is ([a0, b0], [c0], Yes, Yes, complement_is_open_so_compact_subset_is_closed).
Reviewed step-by-step derivation
A compact subset of a Hausdorff space is closed · qf-mod43-compactclosed-finite-closedness-audit · candidate
27. Closedness Uses the Open Complement in the Same Ambient Space
Use complete finite labeled carrier=[a1, b1, c1], topology=[[a1, b1, c1], [a1, b1], [a1, c1], [a1], [b1, c1], [b1], [c1], []] in its original supplied open-set order, and compact_subset K=[a1, b1]. Qualify the topology and every Hausdorff distinct-point pair. Finite K is compact; the general theorem still needs both ambient Hausdorffness and compactness. Give (compact_subset, complement, compact, closed, certificate).
View answer and explanation
([a1, b1], [c1], Yes, Yes, complement_is_open_so_compact_subset_is_closed)
Require the positive Hausdorff instance and verify that X minus K is open. Empty K is also closed. Finite model verification does not prove the general theorem or its converse closed-implies-compact. The exact result, retaining the requested order, is ([a1, b1], [c1], Yes, Yes, complement_is_open_so_compact_subset_is_closed).
Reviewed step-by-step derivation
A compact subset of a Hausdorff space is closed · qf-mod43-compactclosed-finite-closedness-audit · candidate
28. Closedness Uses the Open Complement in the Same Ambient Space
Use complete finite labeled carrier=[a2, b2, c2], topology=[[a2, b2, c2], [a2, b2], [a2, c2], [a2], [b2, c2], [b2], [c2], []] in its original supplied open-set order, and compact_subset K=[a2, b2]. Qualify the topology and every Hausdorff distinct-point pair. Finite K is compact; the general theorem still needs both ambient Hausdorffness and compactness. Give (compact_subset, complement, compact, closed, certificate).
View answer and explanation
([a2, b2], [c2], Yes, Yes, complement_is_open_so_compact_subset_is_closed)
Require the positive Hausdorff instance and verify that X minus K is open. Empty K is also closed. Finite model verification does not prove the general theorem or its converse closed-implies-compact. The exact result, retaining the requested order, is ([a2, b2], [c2], Yes, Yes, complement_is_open_so_compact_subset_is_closed).
Reviewed step-by-step derivation
A compact subset of a Hausdorff space is closed · qf-mod43-compactclosed-finite-closedness-audit · candidate
29. Closedness Uses the Open Complement in the Same Ambient Space
Use complete finite labeled carrier=[a3, b3, c3], topology=[[a3, b3, c3], [a3, b3], [a3, c3], [a3], [b3, c3], [b3], [c3], []] in its original supplied open-set order, and compact_subset K=[a3, b3]. Qualify the topology and every Hausdorff distinct-point pair. Finite K is compact; the general theorem still needs both ambient Hausdorffness and compactness. Give (compact_subset, complement, compact, closed, certificate).
View answer and explanation
([a3, b3], [c3], Yes, Yes, complement_is_open_so_compact_subset_is_closed)
Require the positive Hausdorff instance and verify that X minus K is open. Empty K is also closed. Finite model verification does not prove the general theorem or its converse closed-implies-compact. The exact result, retaining the requested order, is ([a3, b3], [c3], Yes, Yes, complement_is_open_so_compact_subset_is_closed).
Reviewed step-by-step derivation
A compact subset of a Hausdorff space is closed · qf-mod43-compactclosed-finite-closedness-audit · candidate
30. Closedness Uses the Open Complement in the Same Ambient Space
Use complete finite labeled carrier=[a4, b4, c4], topology=[[a4, b4, c4], [a4, b4], [a4, c4], [a4], [b4, c4], [b4], [c4], []] in its original supplied open-set order, and compact_subset K=[a4, b4]. Qualify the topology and every Hausdorff distinct-point pair. Finite K is compact; the general theorem still needs both ambient Hausdorffness and compactness. Give (compact_subset, complement, compact, closed, certificate).
View answer and explanation
([a4, b4], [c4], Yes, Yes, complement_is_open_so_compact_subset_is_closed)
Require the positive Hausdorff instance and verify that X minus K is open. Empty K is also closed. Finite model verification does not prove the general theorem or its converse closed-implies-compact. The exact result, retaining the requested order, is ([a4, b4], [c4], Yes, Yes, complement_is_open_so_compact_subset_is_closed).
Reviewed step-by-step derivation
A compact subset of a Hausdorff space is closed · qf-mod43-compactclosed-finite-closedness-audit · candidate
31. Closedness Uses the Open Complement in the Same Ambient Space
Use complete finite labeled carrier=[a5, b5, c5], topology=[[a5, b5, c5], [a5, b5], [a5, c5], [a5], [b5, c5], [b5], [c5], []] in its original supplied open-set order, and compact_subset K=[a5, b5]. Qualify the topology and every Hausdorff distinct-point pair. Finite K is compact; the general theorem still needs both ambient Hausdorffness and compactness. Give (compact_subset, complement, compact, closed, certificate).
View answer and explanation
([a5, b5], [c5], Yes, Yes, complement_is_open_so_compact_subset_is_closed)
Require the positive Hausdorff instance and verify that X minus K is open. Empty K is also closed. Finite model verification does not prove the general theorem or its converse closed-implies-compact. The exact result, retaining the requested order, is ([a5, b5], [c5], Yes, Yes, complement_is_open_so_compact_subset_is_closed).
Reviewed step-by-step derivation
A compact subset of a Hausdorff space is closed · qf-mod43-compactclosed-finite-closedness-audit · candidate
32. Closedness Uses the Open Complement in the Same Ambient Space
Use complete finite labeled carrier=[a6, b6, c6], topology=[[a6, b6, c6], [a6, b6], [a6, c6], [a6], [b6, c6], [b6], [c6], []] in its original supplied open-set order, and compact_subset K=[a6, b6]. Qualify the topology and every Hausdorff distinct-point pair. Finite K is compact; the general theorem still needs both ambient Hausdorffness and compactness. Give (compact_subset, complement, compact, closed, certificate).
View answer and explanation
([a6, b6], [c6], Yes, Yes, complement_is_open_so_compact_subset_is_closed)
Require the positive Hausdorff instance and verify that X minus K is open. Empty K is also closed. Finite model verification does not prove the general theorem or its converse closed-implies-compact. The exact result, retaining the requested order, is ([a6, b6], [c6], Yes, Yes, complement_is_open_so_compact_subset_is_closed).
Reviewed step-by-step derivation
A compact subset of a Hausdorff space is closed · qf-mod43-compactclosed-finite-closedness-audit · candidate
33. Closedness Uses the Open Complement in the Same Ambient Space
Use complete finite labeled carrier=[a7, b7, c7], topology=[[a7, b7, c7], [a7, b7], [a7, c7], [a7], [b7, c7], [b7], [c7], []] in its original supplied open-set order, and compact_subset K=[a7, b7]. Qualify the topology and every Hausdorff distinct-point pair. Finite K is compact; the general theorem still needs both ambient Hausdorffness and compactness. Give (compact_subset, complement, compact, closed, certificate).
View answer and explanation
([a7, b7], [c7], Yes, Yes, complement_is_open_so_compact_subset_is_closed)
Require the positive Hausdorff instance and verify that X minus K is open. Empty K is also closed. Finite model verification does not prove the general theorem or its converse closed-implies-compact. The exact result, retaining the requested order, is ([a7, b7], [c7], Yes, Yes, complement_is_open_so_compact_subset_is_closed).
Reviewed step-by-step derivation
A compact subset of a Hausdorff space is closed · qf-mod43-compactclosed-finite-closedness-audit · candidate
34. Closedness Uses the Open Complement in the Same Ambient Space
Use complete finite labeled carrier=[a8, b8, c8], topology=[[a8, b8, c8], [a8, b8], [a8, c8], [a8], [b8, c8], [b8], [c8], []] in its original supplied open-set order, and compact_subset K=[a8, b8]. Qualify the topology and every Hausdorff distinct-point pair. Finite K is compact; the general theorem still needs both ambient Hausdorffness and compactness. Give (compact_subset, complement, compact, closed, certificate).
View answer and explanation
([a8, b8], [c8], Yes, Yes, complement_is_open_so_compact_subset_is_closed)
Require the positive Hausdorff instance and verify that X minus K is open. Empty K is also closed. Finite model verification does not prove the general theorem or its converse closed-implies-compact. The exact result, retaining the requested order, is ([a8, b8], [c8], Yes, Yes, complement_is_open_so_compact_subset_is_closed).
Reviewed step-by-step derivation
A compact subset of a Hausdorff space is closed · qf-mod43-compactclosed-finite-closedness-audit · candidate
35. Closedness Uses the Open Complement in the Same Ambient Space
Use complete finite labeled carrier=[a9, b9, c9], topology=[[a9, b9, c9], [a9, b9], [a9, c9], [a9], [b9, c9], [b9], [c9], []] in its original supplied open-set order, and compact_subset K=[a9, b9]. Qualify the topology and every Hausdorff distinct-point pair. Finite K is compact; the general theorem still needs both ambient Hausdorffness and compactness. Give (compact_subset, complement, compact, closed, certificate).
View answer and explanation
([a9, b9], [c9], Yes, Yes, complement_is_open_so_compact_subset_is_closed)
Require the positive Hausdorff instance and verify that X minus K is open. Empty K is also closed. Finite model verification does not prove the general theorem or its converse closed-implies-compact. The exact result, retaining the requested order, is ([a9, b9], [c9], Yes, Yes, complement_is_open_so_compact_subset_is_closed).
Reviewed step-by-step derivation
A compact subset of a Hausdorff space is closed · qf-mod43-compactclosed-finite-closedness-audit · candidate
36. Closedness Uses the Open Complement in the Same Ambient Space
Use complete finite labeled carrier=[a10, b10, c10], topology=[[a10, b10, c10], [a10, b10], [a10, c10], [a10], [b10, c10], [b10], [c10], []] in its original supplied open-set order, and compact_subset K=[a10, b10]. Qualify the topology and every Hausdorff distinct-point pair. Finite K is compact; the general theorem still needs both ambient Hausdorffness and compactness. Give (compact_subset, complement, compact, closed, certificate).
View answer and explanation
([a10, b10], [c10], Yes, Yes, complement_is_open_so_compact_subset_is_closed)
Require the positive Hausdorff instance and verify that X minus K is open. Empty K is also closed. Finite model verification does not prove the general theorem or its converse closed-implies-compact. The exact result, retaining the requested order, is ([a10, b10], [c10], Yes, Yes, complement_is_open_so_compact_subset_is_closed).
Reviewed step-by-step derivation
A compact subset of a Hausdorff space is closed · qf-mod43-compactclosed-finite-closedness-audit · candidate
37. Closedness Uses the Open Complement in the Same Ambient Space
Use complete finite labeled carrier=[a11, b11, c11], topology=[[a11, b11, c11], [a11, b11], [a11, c11], [a11], [b11, c11], [b11], [c11], []] in its original supplied open-set order, and compact_subset K=[a11, b11]. Qualify the topology and every Hausdorff distinct-point pair. Finite K is compact; the general theorem still needs both ambient Hausdorffness and compactness. Give (compact_subset, complement, compact, closed, certificate).
View answer and explanation
([a11, b11], [c11], Yes, Yes, complement_is_open_so_compact_subset_is_closed)
Require the positive Hausdorff instance and verify that X minus K is open. Empty K is also closed. Finite model verification does not prove the general theorem or its converse closed-implies-compact. The exact result, retaining the requested order, is ([a11, b11], [c11], Yes, Yes, complement_is_open_so_compact_subset_is_closed).
Reviewed step-by-step derivation
A compact subset of a Hausdorff space is closed · qf-mod43-compactclosed-finite-closedness-audit · candidate
38. Closedness Uses the Open Complement in the Same Ambient Space
Use complete finite labeled carrier=[a12, b12, c12], topology=[[a12, b12, c12], [a12, b12], [a12, c12], [a12], [b12, c12], [b12], [c12], []] in its original supplied open-set order, and compact_subset K=[a12, b12]. Qualify the topology and every Hausdorff distinct-point pair. Finite K is compact; the general theorem still needs both ambient Hausdorffness and compactness. Give (compact_subset, complement, compact, closed, certificate).
View answer and explanation
([a12, b12], [c12], Yes, Yes, complement_is_open_so_compact_subset_is_closed)
Require the positive Hausdorff instance and verify that X minus K is open. Empty K is also closed. Finite model verification does not prove the general theorem or its converse closed-implies-compact. The exact result, retaining the requested order, is ([a12, b12], [c12], Yes, Yes, complement_is_open_so_compact_subset_is_closed).
Reviewed step-by-step derivation
A compact subset of a Hausdorff space is closed · qf-mod43-compactclosed-finite-closedness-audit · candidate
39. Closedness Uses the Open Complement in the Same Ambient Space
Use complete finite labeled carrier=[a13, b13, c13], topology=[[a13, b13, c13], [a13, b13], [a13, c13], [a13], [b13, c13], [b13], [c13], []] in its original supplied open-set order, and compact_subset K=[a13, b13]. Qualify the topology and every Hausdorff distinct-point pair. Finite K is compact; the general theorem still needs both ambient Hausdorffness and compactness. Give (compact_subset, complement, compact, closed, certificate).
View answer and explanation
([a13, b13], [c13], Yes, Yes, complement_is_open_so_compact_subset_is_closed)
Require the positive Hausdorff instance and verify that X minus K is open. Empty K is also closed. Finite model verification does not prove the general theorem or its converse closed-implies-compact. The exact result, retaining the requested order, is ([a13, b13], [c13], Yes, Yes, complement_is_open_so_compact_subset_is_closed).
Reviewed step-by-step derivation
A compact subset of a Hausdorff space is closed · qf-mod43-compactclosed-finite-closedness-audit · candidate
40. Closedness Uses the Open Complement in the Same Ambient Space
Use complete finite labeled carrier=[a14, b14, c14], topology=[[a14, b14, c14], [a14, b14], [a14, c14], [a14], [b14, c14], [b14], [c14], []] in its original supplied open-set order, and compact_subset K=[a14, b14]. Qualify the topology and every Hausdorff distinct-point pair. Finite K is compact; the general theorem still needs both ambient Hausdorffness and compactness. Give (compact_subset, complement, compact, closed, certificate).
View answer and explanation
([a14, b14], [c14], Yes, Yes, complement_is_open_so_compact_subset_is_closed)
Require the positive Hausdorff instance and verify that X minus K is open. Empty K is also closed. Finite model verification does not prove the general theorem or its converse closed-implies-compact. The exact result, retaining the requested order, is ([a14, b14], [c14], Yes, Yes, complement_is_open_so_compact_subset_is_closed).
Reviewed step-by-step derivation
A compact subset of a Hausdorff space is closed · qf-mod43-compactclosed-finite-closedness-audit · candidate
41. Closedness Uses the Open Complement in the Same Ambient Space
Use complete finite labeled carrier=[a15, b15, c15], topology=[[a15, b15, c15], [a15, b15], [a15, c15], [a15], [b15, c15], [b15], [c15], []] in its original supplied open-set order, and compact_subset K=[a15, b15]. Qualify the topology and every Hausdorff distinct-point pair. Finite K is compact; the general theorem still needs both ambient Hausdorffness and compactness. Give (compact_subset, complement, compact, closed, certificate).
View answer and explanation
([a15, b15], [c15], Yes, Yes, complement_is_open_so_compact_subset_is_closed)
Require the positive Hausdorff instance and verify that X minus K is open. Empty K is also closed. Finite model verification does not prove the general theorem or its converse closed-implies-compact. The exact result, retaining the requested order, is ([a15, b15], [c15], Yes, Yes, complement_is_open_so_compact_subset_is_closed).
Reviewed step-by-step derivation
A compact subset of a Hausdorff space is closed · qf-mod43-compactclosed-finite-closedness-audit · candidate
42. Closedness Uses the Open Complement in the Same Ambient Space
Use complete finite labeled carrier=[a16, b16, c16], topology=[[a16, b16, c16], [a16, b16], [a16, c16], [a16], [b16, c16], [b16], [c16], []] in its original supplied open-set order, and compact_subset K=[a16, b16]. Qualify the topology and every Hausdorff distinct-point pair. Finite K is compact; the general theorem still needs both ambient Hausdorffness and compactness. Give (compact_subset, complement, compact, closed, certificate).
View answer and explanation
([a16, b16], [c16], Yes, Yes, complement_is_open_so_compact_subset_is_closed)
Require the positive Hausdorff instance and verify that X minus K is open. Empty K is also closed. Finite model verification does not prove the general theorem or its converse closed-implies-compact. The exact result, retaining the requested order, is ([a16, b16], [c16], Yes, Yes, complement_is_open_so_compact_subset_is_closed).
Reviewed step-by-step derivation
A compact subset of a Hausdorff space is closed · qf-mod43-compactclosed-finite-closedness-audit · candidate
43. Closedness Uses the Open Complement in the Same Ambient Space
Use complete finite labeled carrier=[a17, b17, c17], topology=[[a17, b17, c17], [a17, b17], [a17, c17], [a17], [b17, c17], [b17], [c17], []] in its original supplied open-set order, and compact_subset K=[a17, b17]. Qualify the topology and every Hausdorff distinct-point pair. Finite K is compact; the general theorem still needs both ambient Hausdorffness and compactness. Give (compact_subset, complement, compact, closed, certificate).
View answer and explanation
([a17, b17], [c17], Yes, Yes, complement_is_open_so_compact_subset_is_closed)
Require the positive Hausdorff instance and verify that X minus K is open. Empty K is also closed. Finite model verification does not prove the general theorem or its converse closed-implies-compact. The exact result, retaining the requested order, is ([a17, b17], [c17], Yes, Yes, complement_is_open_so_compact_subset_is_closed).
Reviewed step-by-step derivation
A compact subset of a Hausdorff space is closed · qf-mod43-compactclosed-finite-closedness-audit · candidate
44. Closedness Uses the Open Complement in the Same Ambient Space
Use complete finite labeled carrier=[a18, b18, c18], topology=[[a18, b18, c18], [a18, b18], [a18, c18], [a18], [b18, c18], [b18], [c18], []] in its original supplied open-set order, and compact_subset K=[a18, b18]. Qualify the topology and every Hausdorff distinct-point pair. Finite K is compact; the general theorem still needs both ambient Hausdorffness and compactness. Give (compact_subset, complement, compact, closed, certificate).
View answer and explanation
([a18, b18], [c18], Yes, Yes, complement_is_open_so_compact_subset_is_closed)
Require the positive Hausdorff instance and verify that X minus K is open. Empty K is also closed. Finite model verification does not prove the general theorem or its converse closed-implies-compact. The exact result, retaining the requested order, is ([a18, b18], [c18], Yes, Yes, complement_is_open_so_compact_subset_is_closed).
Reviewed step-by-step derivation
A compact subset of a Hausdorff space is closed · qf-mod43-compactclosed-finite-closedness-audit · candidate
45. Closedness Uses the Open Complement in the Same Ambient Space
Use complete finite labeled carrier=[a19, b19, c19], topology=[[a19, b19, c19], [a19, b19], [a19, c19], [a19], [b19, c19], [b19], [c19], []] in its original supplied open-set order, and compact_subset K=[a19, b19]. Qualify the topology and every Hausdorff distinct-point pair. Finite K is compact; the general theorem still needs both ambient Hausdorffness and compactness. Give (compact_subset, complement, compact, closed, certificate).
View answer and explanation
([a19, b19], [c19], Yes, Yes, complement_is_open_so_compact_subset_is_closed)
Require the positive Hausdorff instance and verify that X minus K is open. Empty K is also closed. Finite model verification does not prove the general theorem or its converse closed-implies-compact. The exact result, retaining the requested order, is ([a19, b19], [c19], Yes, Yes, complement_is_open_so_compact_subset_is_closed).
Reviewed step-by-step derivation
A compact subset of a Hausdorff space is closed · qf-mod43-compactclosed-finite-closedness-audit · candidate
46. Closedness Uses the Open Complement in the Same Ambient Space
Use complete finite labeled carrier=[a20, b20, c20], topology=[[a20, b20, c20], [a20, b20], [a20, c20], [a20], [b20, c20], [b20], [c20], []] in its original supplied open-set order, and compact_subset K=[a20, b20]. Qualify the topology and every Hausdorff distinct-point pair. Finite K is compact; the general theorem still needs both ambient Hausdorffness and compactness. Give (compact_subset, complement, compact, closed, certificate).
View answer and explanation
([a20, b20], [c20], Yes, Yes, complement_is_open_so_compact_subset_is_closed)
Require the positive Hausdorff instance and verify that X minus K is open. Empty K is also closed. Finite model verification does not prove the general theorem or its converse closed-implies-compact. The exact result, retaining the requested order, is ([a20, b20], [c20], Yes, Yes, complement_is_open_so_compact_subset_is_closed).
Reviewed step-by-step derivation
A compact subset of a Hausdorff space is closed · qf-mod43-compactclosed-finite-closedness-audit · candidate
47. Closedness Uses the Open Complement in the Same Ambient Space
Use complete finite labeled carrier=[a21, b21, c21], topology=[[a21, b21, c21], [a21, b21], [a21, c21], [a21], [b21, c21], [b21], [c21], []] in its original supplied open-set order, and compact_subset K=[a21, b21]. Qualify the topology and every Hausdorff distinct-point pair. Finite K is compact; the general theorem still needs both ambient Hausdorffness and compactness. Give (compact_subset, complement, compact, closed, certificate).
View answer and explanation
([a21, b21], [c21], Yes, Yes, complement_is_open_so_compact_subset_is_closed)
Require the positive Hausdorff instance and verify that X minus K is open. Empty K is also closed. Finite model verification does not prove the general theorem or its converse closed-implies-compact. The exact result, retaining the requested order, is ([a21, b21], [c21], Yes, Yes, complement_is_open_so_compact_subset_is_closed).
Reviewed step-by-step derivation
A compact subset of a Hausdorff space is closed · qf-mod43-compactclosed-finite-closedness-audit · candidate
48. Closedness Uses the Open Complement in the Same Ambient Space
Use complete finite labeled carrier=[a22, b22, c22], topology=[[a22, b22, c22], [a22, b22], [a22, c22], [a22], [b22, c22], [b22], [c22], []] in its original supplied open-set order, and compact_subset K=[a22, b22]. Qualify the topology and every Hausdorff distinct-point pair. Finite K is compact; the general theorem still needs both ambient Hausdorffness and compactness. Give (compact_subset, complement, compact, closed, certificate).
View answer and explanation
([a22, b22], [c22], Yes, Yes, complement_is_open_so_compact_subset_is_closed)
Require the positive Hausdorff instance and verify that X minus K is open. Empty K is also closed. Finite model verification does not prove the general theorem or its converse closed-implies-compact. The exact result, retaining the requested order, is ([a22, b22], [c22], Yes, Yes, complement_is_open_so_compact_subset_is_closed).
Reviewed step-by-step derivation
A compact subset of a Hausdorff space is closed · qf-mod43-compactclosed-finite-closedness-audit · candidate
49. Closedness Uses the Open Complement in the Same Ambient Space
Use complete finite labeled carrier=[a23, b23, c23], topology=[[a23, b23, c23], [a23, b23], [a23, c23], [a23], [b23, c23], [b23], [c23], []] in its original supplied open-set order, and compact_subset K=[a23, b23]. Qualify the topology and every Hausdorff distinct-point pair. Finite K is compact; the general theorem still needs both ambient Hausdorffness and compactness. Give (compact_subset, complement, compact, closed, certificate).
View answer and explanation
([a23, b23], [c23], Yes, Yes, complement_is_open_so_compact_subset_is_closed)
Require the positive Hausdorff instance and verify that X minus K is open. Empty K is also closed. Finite model verification does not prove the general theorem or its converse closed-implies-compact. The exact result, retaining the requested order, is ([a23, b23], [c23], Yes, Yes, complement_is_open_so_compact_subset_is_closed).
Reviewed step-by-step derivation
A compact subset of a Hausdorff space is closed · qf-mod43-compactclosed-finite-closedness-audit · candidate
50. Closedness Uses the Open Complement in the Same Ambient Space
Use complete finite labeled carrier=[a24, b24, c24], topology=[[a24, b24, c24], [a24, b24], [a24, c24], [a24], [b24, c24], [b24], [c24], []] in its original supplied open-set order, and compact_subset K=[a24, b24]. Qualify the topology and every Hausdorff distinct-point pair. Finite K is compact; the general theorem still needs both ambient Hausdorffness and compactness. Give (compact_subset, complement, compact, closed, certificate).
View answer and explanation
([a24, b24], [c24], Yes, Yes, complement_is_open_so_compact_subset_is_closed)
Require the positive Hausdorff instance and verify that X minus K is open. Empty K is also closed. Finite model verification does not prove the general theorem or its converse closed-implies-compact. The exact result, retaining the requested order, is ([a24, b24], [c24], Yes, Yes, complement_is_open_so_compact_subset_is_closed).
Reviewed step-by-step derivation
A compact subset of a Hausdorff space is closed · qf-mod43-compactclosed-finite-closedness-audit · candidate
51. Finite Intersections Separate an Exterior Point from the Whole Compact Subset
Use complete finite labeled carrier=[a0, b0, c0], topology=[[a0, b0, c0], [a0, b0], [a0, c0], [a0], [b0, c0], [b0], [c0], []] in its original supplied open-set order, and compact_subset K=[a0, b0]. Qualify the topology and every Hausdorff distinct-point pair. Finite K is compact; the general theorem still needs both ambient Hausdorffness and compactness. Exterior point x=c0 must lie outside K. Give (exterior_point, separation_rows, exterior_neighborhood, cover_union, covers_K, neighborhood_disjoint_from_K, neighborhood_open, certificate).
cover_union may extend beyond K. The exterior neighborhood is a finite intersection and must be open and disjoint from K. No minimum or unique subcover is asserted.
View answer and explanation
(c0, [[a0, [[b0, c0], [a0]]], [b0, [[a0, c0], [b0]]]], [c0], [a0, b0], Yes, Yes, Yes, finite_subcover_and_finite_intersection_separate_x_from_K)
For each y∈K choose disjoint U_y containing x and V_y containing y in original open-list order. The V_y union covers K but may exceed K; intersect the finitely many U_y to obtain an open neighborhood disjoint from K. For empty K the empty intersection is X. No minimum or unique subcover is asserted; the general proof uses compactness to obtain a finite subcover. The exact result, retaining the requested order, is (c0, [[a0, [[b0, c0], [a0]]], [b0, [[a0, c0], [b0]]]], [c0], [a0, b0], Yes, Yes, Yes, finite_subcover_and_finite_intersection_separate_x_from_K).
Reviewed step-by-step derivation
A compact subset of a Hausdorff space is closed · qf-mod43-compactclosed-neighborhood-subcover-certificate · candidate
52. Finite Intersections Separate an Exterior Point from the Whole Compact Subset
Use complete finite labeled carrier=[a1, b1, c1], topology=[[a1, b1, c1], [a1, b1], [a1, c1], [a1], [b1, c1], [b1], [c1], []] in its original supplied open-set order, and compact_subset K=[a1, b1]. Qualify the topology and every Hausdorff distinct-point pair. Finite K is compact; the general theorem still needs both ambient Hausdorffness and compactness. Exterior point x=c1 must lie outside K. Give (exterior_point, separation_rows, exterior_neighborhood, cover_union, covers_K, neighborhood_disjoint_from_K, neighborhood_open, certificate).
cover_union may extend beyond K. The exterior neighborhood is a finite intersection and must be open and disjoint from K. No minimum or unique subcover is asserted.
View answer and explanation
(c1, [[a1, [[b1, c1], [a1]]], [b1, [[a1, c1], [b1]]]], [c1], [a1, b1], Yes, Yes, Yes, finite_subcover_and_finite_intersection_separate_x_from_K)
For each y∈K choose disjoint U_y containing x and V_y containing y in original open-list order. The V_y union covers K but may exceed K; intersect the finitely many U_y to obtain an open neighborhood disjoint from K. For empty K the empty intersection is X. No minimum or unique subcover is asserted; the general proof uses compactness to obtain a finite subcover. The exact result, retaining the requested order, is (c1, [[a1, [[b1, c1], [a1]]], [b1, [[a1, c1], [b1]]]], [c1], [a1, b1], Yes, Yes, Yes, finite_subcover_and_finite_intersection_separate_x_from_K).
Reviewed step-by-step derivation
A compact subset of a Hausdorff space is closed · qf-mod43-compactclosed-neighborhood-subcover-certificate · candidate
53. Finite Intersections Separate an Exterior Point from the Whole Compact Subset
Use complete finite labeled carrier=[a2, b2, c2], topology=[[a2, b2, c2], [a2, b2], [a2, c2], [a2], [b2, c2], [b2], [c2], []] in its original supplied open-set order, and compact_subset K=[a2, b2]. Qualify the topology and every Hausdorff distinct-point pair. Finite K is compact; the general theorem still needs both ambient Hausdorffness and compactness. Exterior point x=c2 must lie outside K. Give (exterior_point, separation_rows, exterior_neighborhood, cover_union, covers_K, neighborhood_disjoint_from_K, neighborhood_open, certificate).
cover_union may extend beyond K. The exterior neighborhood is a finite intersection and must be open and disjoint from K. No minimum or unique subcover is asserted.
View answer and explanation
(c2, [[a2, [[b2, c2], [a2]]], [b2, [[a2, c2], [b2]]]], [c2], [a2, b2], Yes, Yes, Yes, finite_subcover_and_finite_intersection_separate_x_from_K)
For each y∈K choose disjoint U_y containing x and V_y containing y in original open-list order. The V_y union covers K but may exceed K; intersect the finitely many U_y to obtain an open neighborhood disjoint from K. For empty K the empty intersection is X. No minimum or unique subcover is asserted; the general proof uses compactness to obtain a finite subcover. The exact result, retaining the requested order, is (c2, [[a2, [[b2, c2], [a2]]], [b2, [[a2, c2], [b2]]]], [c2], [a2, b2], Yes, Yes, Yes, finite_subcover_and_finite_intersection_separate_x_from_K).
Reviewed step-by-step derivation
A compact subset of a Hausdorff space is closed · qf-mod43-compactclosed-neighborhood-subcover-certificate · candidate
54. Finite Intersections Separate an Exterior Point from the Whole Compact Subset
Use complete finite labeled carrier=[a3, b3, c3], topology=[[a3, b3, c3], [a3, b3], [a3, c3], [a3], [b3, c3], [b3], [c3], []] in its original supplied open-set order, and compact_subset K=[a3, b3]. Qualify the topology and every Hausdorff distinct-point pair. Finite K is compact; the general theorem still needs both ambient Hausdorffness and compactness. Exterior point x=c3 must lie outside K. Give (exterior_point, separation_rows, exterior_neighborhood, cover_union, covers_K, neighborhood_disjoint_from_K, neighborhood_open, certificate).
cover_union may extend beyond K. The exterior neighborhood is a finite intersection and must be open and disjoint from K. No minimum or unique subcover is asserted.
View answer and explanation
(c3, [[a3, [[b3, c3], [a3]]], [b3, [[a3, c3], [b3]]]], [c3], [a3, b3], Yes, Yes, Yes, finite_subcover_and_finite_intersection_separate_x_from_K)
For each y∈K choose disjoint U_y containing x and V_y containing y in original open-list order. The V_y union covers K but may exceed K; intersect the finitely many U_y to obtain an open neighborhood disjoint from K. For empty K the empty intersection is X. No minimum or unique subcover is asserted; the general proof uses compactness to obtain a finite subcover. The exact result, retaining the requested order, is (c3, [[a3, [[b3, c3], [a3]]], [b3, [[a3, c3], [b3]]]], [c3], [a3, b3], Yes, Yes, Yes, finite_subcover_and_finite_intersection_separate_x_from_K).
Reviewed step-by-step derivation
A compact subset of a Hausdorff space is closed · qf-mod43-compactclosed-neighborhood-subcover-certificate · candidate
55. Finite Intersections Separate an Exterior Point from the Whole Compact Subset
Use complete finite labeled carrier=[a4, b4, c4], topology=[[a4, b4, c4], [a4, b4], [a4, c4], [a4], [b4, c4], [b4], [c4], []] in its original supplied open-set order, and compact_subset K=[a4, b4]. Qualify the topology and every Hausdorff distinct-point pair. Finite K is compact; the general theorem still needs both ambient Hausdorffness and compactness. Exterior point x=c4 must lie outside K. Give (exterior_point, separation_rows, exterior_neighborhood, cover_union, covers_K, neighborhood_disjoint_from_K, neighborhood_open, certificate).
cover_union may extend beyond K. The exterior neighborhood is a finite intersection and must be open and disjoint from K. No minimum or unique subcover is asserted.
View answer and explanation
(c4, [[a4, [[b4, c4], [a4]]], [b4, [[a4, c4], [b4]]]], [c4], [a4, b4], Yes, Yes, Yes, finite_subcover_and_finite_intersection_separate_x_from_K)
For each y∈K choose disjoint U_y containing x and V_y containing y in original open-list order. The V_y union covers K but may exceed K; intersect the finitely many U_y to obtain an open neighborhood disjoint from K. For empty K the empty intersection is X. No minimum or unique subcover is asserted; the general proof uses compactness to obtain a finite subcover. The exact result, retaining the requested order, is (c4, [[a4, [[b4, c4], [a4]]], [b4, [[a4, c4], [b4]]]], [c4], [a4, b4], Yes, Yes, Yes, finite_subcover_and_finite_intersection_separate_x_from_K).
Reviewed step-by-step derivation
A compact subset of a Hausdorff space is closed · qf-mod43-compactclosed-neighborhood-subcover-certificate · candidate
56. Finite Intersections Separate an Exterior Point from the Whole Compact Subset
Use complete finite labeled carrier=[a5, b5, c5], topology=[[a5, b5, c5], [a5, b5], [a5, c5], [a5], [b5, c5], [b5], [c5], []] in its original supplied open-set order, and compact_subset K=[a5, b5]. Qualify the topology and every Hausdorff distinct-point pair. Finite K is compact; the general theorem still needs both ambient Hausdorffness and compactness. Exterior point x=c5 must lie outside K. Give (exterior_point, separation_rows, exterior_neighborhood, cover_union, covers_K, neighborhood_disjoint_from_K, neighborhood_open, certificate).
cover_union may extend beyond K. The exterior neighborhood is a finite intersection and must be open and disjoint from K. No minimum or unique subcover is asserted.
View answer and explanation
(c5, [[a5, [[b5, c5], [a5]]], [b5, [[a5, c5], [b5]]]], [c5], [a5, b5], Yes, Yes, Yes, finite_subcover_and_finite_intersection_separate_x_from_K)
For each y∈K choose disjoint U_y containing x and V_y containing y in original open-list order. The V_y union covers K but may exceed K; intersect the finitely many U_y to obtain an open neighborhood disjoint from K. For empty K the empty intersection is X. No minimum or unique subcover is asserted; the general proof uses compactness to obtain a finite subcover. The exact result, retaining the requested order, is (c5, [[a5, [[b5, c5], [a5]]], [b5, [[a5, c5], [b5]]]], [c5], [a5, b5], Yes, Yes, Yes, finite_subcover_and_finite_intersection_separate_x_from_K).
Reviewed step-by-step derivation
A compact subset of a Hausdorff space is closed · qf-mod43-compactclosed-neighborhood-subcover-certificate · candidate
57. Finite Intersections Separate an Exterior Point from the Whole Compact Subset
Use complete finite labeled carrier=[a6, b6, c6], topology=[[a6, b6, c6], [a6, b6], [a6, c6], [a6], [b6, c6], [b6], [c6], []] in its original supplied open-set order, and compact_subset K=[a6, b6]. Qualify the topology and every Hausdorff distinct-point pair. Finite K is compact; the general theorem still needs both ambient Hausdorffness and compactness. Exterior point x=c6 must lie outside K. Give (exterior_point, separation_rows, exterior_neighborhood, cover_union, covers_K, neighborhood_disjoint_from_K, neighborhood_open, certificate).
cover_union may extend beyond K. The exterior neighborhood is a finite intersection and must be open and disjoint from K. No minimum or unique subcover is asserted.
View answer and explanation
(c6, [[a6, [[b6, c6], [a6]]], [b6, [[a6, c6], [b6]]]], [c6], [a6, b6], Yes, Yes, Yes, finite_subcover_and_finite_intersection_separate_x_from_K)
For each y∈K choose disjoint U_y containing x and V_y containing y in original open-list order. The V_y union covers K but may exceed K; intersect the finitely many U_y to obtain an open neighborhood disjoint from K. For empty K the empty intersection is X. No minimum or unique subcover is asserted; the general proof uses compactness to obtain a finite subcover. The exact result, retaining the requested order, is (c6, [[a6, [[b6, c6], [a6]]], [b6, [[a6, c6], [b6]]]], [c6], [a6, b6], Yes, Yes, Yes, finite_subcover_and_finite_intersection_separate_x_from_K).
Reviewed step-by-step derivation
A compact subset of a Hausdorff space is closed · qf-mod43-compactclosed-neighborhood-subcover-certificate · candidate
58. Finite Intersections Separate an Exterior Point from the Whole Compact Subset
Use complete finite labeled carrier=[a7, b7, c7], topology=[[a7, b7, c7], [a7, b7], [a7, c7], [a7], [b7, c7], [b7], [c7], []] in its original supplied open-set order, and compact_subset K=[a7, b7]. Qualify the topology and every Hausdorff distinct-point pair. Finite K is compact; the general theorem still needs both ambient Hausdorffness and compactness. Exterior point x=c7 must lie outside K. Give (exterior_point, separation_rows, exterior_neighborhood, cover_union, covers_K, neighborhood_disjoint_from_K, neighborhood_open, certificate).
cover_union may extend beyond K. The exterior neighborhood is a finite intersection and must be open and disjoint from K. No minimum or unique subcover is asserted.
View answer and explanation
(c7, [[a7, [[b7, c7], [a7]]], [b7, [[a7, c7], [b7]]]], [c7], [a7, b7], Yes, Yes, Yes, finite_subcover_and_finite_intersection_separate_x_from_K)
For each y∈K choose disjoint U_y containing x and V_y containing y in original open-list order. The V_y union covers K but may exceed K; intersect the finitely many U_y to obtain an open neighborhood disjoint from K. For empty K the empty intersection is X. No minimum or unique subcover is asserted; the general proof uses compactness to obtain a finite subcover. The exact result, retaining the requested order, is (c7, [[a7, [[b7, c7], [a7]]], [b7, [[a7, c7], [b7]]]], [c7], [a7, b7], Yes, Yes, Yes, finite_subcover_and_finite_intersection_separate_x_from_K).
Reviewed step-by-step derivation
A compact subset of a Hausdorff space is closed · qf-mod43-compactclosed-neighborhood-subcover-certificate · candidate
59. Finite Intersections Separate an Exterior Point from the Whole Compact Subset
Use complete finite labeled carrier=[a8, b8, c8], topology=[[a8, b8, c8], [a8, b8], [a8, c8], [a8], [b8, c8], [b8], [c8], []] in its original supplied open-set order, and compact_subset K=[a8, b8]. Qualify the topology and every Hausdorff distinct-point pair. Finite K is compact; the general theorem still needs both ambient Hausdorffness and compactness. Exterior point x=c8 must lie outside K. Give (exterior_point, separation_rows, exterior_neighborhood, cover_union, covers_K, neighborhood_disjoint_from_K, neighborhood_open, certificate).
cover_union may extend beyond K. The exterior neighborhood is a finite intersection and must be open and disjoint from K. No minimum or unique subcover is asserted.
View answer and explanation
(c8, [[a8, [[b8, c8], [a8]]], [b8, [[a8, c8], [b8]]]], [c8], [a8, b8], Yes, Yes, Yes, finite_subcover_and_finite_intersection_separate_x_from_K)
For each y∈K choose disjoint U_y containing x and V_y containing y in original open-list order. The V_y union covers K but may exceed K; intersect the finitely many U_y to obtain an open neighborhood disjoint from K. For empty K the empty intersection is X. No minimum or unique subcover is asserted; the general proof uses compactness to obtain a finite subcover. The exact result, retaining the requested order, is (c8, [[a8, [[b8, c8], [a8]]], [b8, [[a8, c8], [b8]]]], [c8], [a8, b8], Yes, Yes, Yes, finite_subcover_and_finite_intersection_separate_x_from_K).
Reviewed step-by-step derivation
A compact subset of a Hausdorff space is closed · qf-mod43-compactclosed-neighborhood-subcover-certificate · candidate
60. Finite Intersections Separate an Exterior Point from the Whole Compact Subset
Use complete finite labeled carrier=[a9, b9, c9], topology=[[a9, b9, c9], [a9, b9], [a9, c9], [a9], [b9, c9], [b9], [c9], []] in its original supplied open-set order, and compact_subset K=[a9, b9]. Qualify the topology and every Hausdorff distinct-point pair. Finite K is compact; the general theorem still needs both ambient Hausdorffness and compactness. Exterior point x=c9 must lie outside K. Give (exterior_point, separation_rows, exterior_neighborhood, cover_union, covers_K, neighborhood_disjoint_from_K, neighborhood_open, certificate).
cover_union may extend beyond K. The exterior neighborhood is a finite intersection and must be open and disjoint from K. No minimum or unique subcover is asserted.
View answer and explanation
(c9, [[a9, [[b9, c9], [a9]]], [b9, [[a9, c9], [b9]]]], [c9], [a9, b9], Yes, Yes, Yes, finite_subcover_and_finite_intersection_separate_x_from_K)
For each y∈K choose disjoint U_y containing x and V_y containing y in original open-list order. The V_y union covers K but may exceed K; intersect the finitely many U_y to obtain an open neighborhood disjoint from K. For empty K the empty intersection is X. No minimum or unique subcover is asserted; the general proof uses compactness to obtain a finite subcover. The exact result, retaining the requested order, is (c9, [[a9, [[b9, c9], [a9]]], [b9, [[a9, c9], [b9]]]], [c9], [a9, b9], Yes, Yes, Yes, finite_subcover_and_finite_intersection_separate_x_from_K).
Reviewed step-by-step derivation
A compact subset of a Hausdorff space is closed · qf-mod43-compactclosed-neighborhood-subcover-certificate · candidate
61. Finite Intersections Separate an Exterior Point from the Whole Compact Subset
Use complete finite labeled carrier=[a10, b10, c10], topology=[[a10, b10, c10], [a10, b10], [a10, c10], [a10], [b10, c10], [b10], [c10], []] in its original supplied open-set order, and compact_subset K=[a10, b10]. Qualify the topology and every Hausdorff distinct-point pair. Finite K is compact; the general theorem still needs both ambient Hausdorffness and compactness. Exterior point x=c10 must lie outside K. Give (exterior_point, separation_rows, exterior_neighborhood, cover_union, covers_K, neighborhood_disjoint_from_K, neighborhood_open, certificate).
cover_union may extend beyond K. The exterior neighborhood is a finite intersection and must be open and disjoint from K. No minimum or unique subcover is asserted.
View answer and explanation
(c10, [[a10, [[b10, c10], [a10]]], [b10, [[a10, c10], [b10]]]], [c10], [a10, b10], Yes, Yes, Yes, finite_subcover_and_finite_intersection_separate_x_from_K)
For each y∈K choose disjoint U_y containing x and V_y containing y in original open-list order. The V_y union covers K but may exceed K; intersect the finitely many U_y to obtain an open neighborhood disjoint from K. For empty K the empty intersection is X. No minimum or unique subcover is asserted; the general proof uses compactness to obtain a finite subcover. The exact result, retaining the requested order, is (c10, [[a10, [[b10, c10], [a10]]], [b10, [[a10, c10], [b10]]]], [c10], [a10, b10], Yes, Yes, Yes, finite_subcover_and_finite_intersection_separate_x_from_K).
Reviewed step-by-step derivation
A compact subset of a Hausdorff space is closed · qf-mod43-compactclosed-neighborhood-subcover-certificate · candidate
62. Finite Intersections Separate an Exterior Point from the Whole Compact Subset
Use complete finite labeled carrier=[a11, b11, c11], topology=[[a11, b11, c11], [a11, b11], [a11, c11], [a11], [b11, c11], [b11], [c11], []] in its original supplied open-set order, and compact_subset K=[a11, b11]. Qualify the topology and every Hausdorff distinct-point pair. Finite K is compact; the general theorem still needs both ambient Hausdorffness and compactness. Exterior point x=c11 must lie outside K. Give (exterior_point, separation_rows, exterior_neighborhood, cover_union, covers_K, neighborhood_disjoint_from_K, neighborhood_open, certificate).
cover_union may extend beyond K. The exterior neighborhood is a finite intersection and must be open and disjoint from K. No minimum or unique subcover is asserted.
View answer and explanation
(c11, [[a11, [[b11, c11], [a11]]], [b11, [[a11, c11], [b11]]]], [c11], [a11, b11], Yes, Yes, Yes, finite_subcover_and_finite_intersection_separate_x_from_K)
For each y∈K choose disjoint U_y containing x and V_y containing y in original open-list order. The V_y union covers K but may exceed K; intersect the finitely many U_y to obtain an open neighborhood disjoint from K. For empty K the empty intersection is X. No minimum or unique subcover is asserted; the general proof uses compactness to obtain a finite subcover. The exact result, retaining the requested order, is (c11, [[a11, [[b11, c11], [a11]]], [b11, [[a11, c11], [b11]]]], [c11], [a11, b11], Yes, Yes, Yes, finite_subcover_and_finite_intersection_separate_x_from_K).
Reviewed step-by-step derivation
A compact subset of a Hausdorff space is closed · qf-mod43-compactclosed-neighborhood-subcover-certificate · candidate
63. Finite Intersections Separate an Exterior Point from the Whole Compact Subset
Use complete finite labeled carrier=[a12, b12, c12], topology=[[a12, b12, c12], [a12, b12], [a12, c12], [a12], [b12, c12], [b12], [c12], []] in its original supplied open-set order, and compact_subset K=[a12, b12]. Qualify the topology and every Hausdorff distinct-point pair. Finite K is compact; the general theorem still needs both ambient Hausdorffness and compactness. Exterior point x=c12 must lie outside K. Give (exterior_point, separation_rows, exterior_neighborhood, cover_union, covers_K, neighborhood_disjoint_from_K, neighborhood_open, certificate).
cover_union may extend beyond K. The exterior neighborhood is a finite intersection and must be open and disjoint from K. No minimum or unique subcover is asserted.
View answer and explanation
(c12, [[a12, [[b12, c12], [a12]]], [b12, [[a12, c12], [b12]]]], [c12], [a12, b12], Yes, Yes, Yes, finite_subcover_and_finite_intersection_separate_x_from_K)
For each y∈K choose disjoint U_y containing x and V_y containing y in original open-list order. The V_y union covers K but may exceed K; intersect the finitely many U_y to obtain an open neighborhood disjoint from K. For empty K the empty intersection is X. No minimum or unique subcover is asserted; the general proof uses compactness to obtain a finite subcover. The exact result, retaining the requested order, is (c12, [[a12, [[b12, c12], [a12]]], [b12, [[a12, c12], [b12]]]], [c12], [a12, b12], Yes, Yes, Yes, finite_subcover_and_finite_intersection_separate_x_from_K).
Reviewed step-by-step derivation
A compact subset of a Hausdorff space is closed · qf-mod43-compactclosed-neighborhood-subcover-certificate · candidate
64. Finite Intersections Separate an Exterior Point from the Whole Compact Subset
Use complete finite labeled carrier=[a13, b13, c13], topology=[[a13, b13, c13], [a13, b13], [a13, c13], [a13], [b13, c13], [b13], [c13], []] in its original supplied open-set order, and compact_subset K=[a13, b13]. Qualify the topology and every Hausdorff distinct-point pair. Finite K is compact; the general theorem still needs both ambient Hausdorffness and compactness. Exterior point x=c13 must lie outside K. Give (exterior_point, separation_rows, exterior_neighborhood, cover_union, covers_K, neighborhood_disjoint_from_K, neighborhood_open, certificate).
cover_union may extend beyond K. The exterior neighborhood is a finite intersection and must be open and disjoint from K. No minimum or unique subcover is asserted.
View answer and explanation
(c13, [[a13, [[b13, c13], [a13]]], [b13, [[a13, c13], [b13]]]], [c13], [a13, b13], Yes, Yes, Yes, finite_subcover_and_finite_intersection_separate_x_from_K)
For each y∈K choose disjoint U_y containing x and V_y containing y in original open-list order. The V_y union covers K but may exceed K; intersect the finitely many U_y to obtain an open neighborhood disjoint from K. For empty K the empty intersection is X. No minimum or unique subcover is asserted; the general proof uses compactness to obtain a finite subcover. The exact result, retaining the requested order, is (c13, [[a13, [[b13, c13], [a13]]], [b13, [[a13, c13], [b13]]]], [c13], [a13, b13], Yes, Yes, Yes, finite_subcover_and_finite_intersection_separate_x_from_K).
Reviewed step-by-step derivation
A compact subset of a Hausdorff space is closed · qf-mod43-compactclosed-neighborhood-subcover-certificate · candidate
65. Finite Intersections Separate an Exterior Point from the Whole Compact Subset
Use complete finite labeled carrier=[a14, b14, c14], topology=[[a14, b14, c14], [a14, b14], [a14, c14], [a14], [b14, c14], [b14], [c14], []] in its original supplied open-set order, and compact_subset K=[a14, b14]. Qualify the topology and every Hausdorff distinct-point pair. Finite K is compact; the general theorem still needs both ambient Hausdorffness and compactness. Exterior point x=c14 must lie outside K. Give (exterior_point, separation_rows, exterior_neighborhood, cover_union, covers_K, neighborhood_disjoint_from_K, neighborhood_open, certificate).
cover_union may extend beyond K. The exterior neighborhood is a finite intersection and must be open and disjoint from K. No minimum or unique subcover is asserted.
View answer and explanation
(c14, [[a14, [[b14, c14], [a14]]], [b14, [[a14, c14], [b14]]]], [c14], [a14, b14], Yes, Yes, Yes, finite_subcover_and_finite_intersection_separate_x_from_K)
For each y∈K choose disjoint U_y containing x and V_y containing y in original open-list order. The V_y union covers K but may exceed K; intersect the finitely many U_y to obtain an open neighborhood disjoint from K. For empty K the empty intersection is X. No minimum or unique subcover is asserted; the general proof uses compactness to obtain a finite subcover. The exact result, retaining the requested order, is (c14, [[a14, [[b14, c14], [a14]]], [b14, [[a14, c14], [b14]]]], [c14], [a14, b14], Yes, Yes, Yes, finite_subcover_and_finite_intersection_separate_x_from_K).
Reviewed step-by-step derivation
A compact subset of a Hausdorff space is closed · qf-mod43-compactclosed-neighborhood-subcover-certificate · candidate
66. Finite Intersections Separate an Exterior Point from the Whole Compact Subset
Use complete finite labeled carrier=[a15, b15, c15], topology=[[a15, b15, c15], [a15, b15], [a15, c15], [a15], [b15, c15], [b15], [c15], []] in its original supplied open-set order, and compact_subset K=[a15, b15]. Qualify the topology and every Hausdorff distinct-point pair. Finite K is compact; the general theorem still needs both ambient Hausdorffness and compactness. Exterior point x=c15 must lie outside K. Give (exterior_point, separation_rows, exterior_neighborhood, cover_union, covers_K, neighborhood_disjoint_from_K, neighborhood_open, certificate).
cover_union may extend beyond K. The exterior neighborhood is a finite intersection and must be open and disjoint from K. No minimum or unique subcover is asserted.
View answer and explanation
(c15, [[a15, [[b15, c15], [a15]]], [b15, [[a15, c15], [b15]]]], [c15], [a15, b15], Yes, Yes, Yes, finite_subcover_and_finite_intersection_separate_x_from_K)
For each y∈K choose disjoint U_y containing x and V_y containing y in original open-list order. The V_y union covers K but may exceed K; intersect the finitely many U_y to obtain an open neighborhood disjoint from K. For empty K the empty intersection is X. No minimum or unique subcover is asserted; the general proof uses compactness to obtain a finite subcover. The exact result, retaining the requested order, is (c15, [[a15, [[b15, c15], [a15]]], [b15, [[a15, c15], [b15]]]], [c15], [a15, b15], Yes, Yes, Yes, finite_subcover_and_finite_intersection_separate_x_from_K).
Reviewed step-by-step derivation
A compact subset of a Hausdorff space is closed · qf-mod43-compactclosed-neighborhood-subcover-certificate · candidate
67. Finite Intersections Separate an Exterior Point from the Whole Compact Subset
Use complete finite labeled carrier=[a16, b16, c16], topology=[[a16, b16, c16], [a16, b16], [a16, c16], [a16], [b16, c16], [b16], [c16], []] in its original supplied open-set order, and compact_subset K=[a16, b16]. Qualify the topology and every Hausdorff distinct-point pair. Finite K is compact; the general theorem still needs both ambient Hausdorffness and compactness. Exterior point x=c16 must lie outside K. Give (exterior_point, separation_rows, exterior_neighborhood, cover_union, covers_K, neighborhood_disjoint_from_K, neighborhood_open, certificate).
cover_union may extend beyond K. The exterior neighborhood is a finite intersection and must be open and disjoint from K. No minimum or unique subcover is asserted.
View answer and explanation
(c16, [[a16, [[b16, c16], [a16]]], [b16, [[a16, c16], [b16]]]], [c16], [a16, b16], Yes, Yes, Yes, finite_subcover_and_finite_intersection_separate_x_from_K)
For each y∈K choose disjoint U_y containing x and V_y containing y in original open-list order. The V_y union covers K but may exceed K; intersect the finitely many U_y to obtain an open neighborhood disjoint from K. For empty K the empty intersection is X. No minimum or unique subcover is asserted; the general proof uses compactness to obtain a finite subcover. The exact result, retaining the requested order, is (c16, [[a16, [[b16, c16], [a16]]], [b16, [[a16, c16], [b16]]]], [c16], [a16, b16], Yes, Yes, Yes, finite_subcover_and_finite_intersection_separate_x_from_K).
Reviewed step-by-step derivation
A compact subset of a Hausdorff space is closed · qf-mod43-compactclosed-neighborhood-subcover-certificate · candidate
68. Finite Intersections Separate an Exterior Point from the Whole Compact Subset
Use complete finite labeled carrier=[a17, b17, c17], topology=[[a17, b17, c17], [a17, b17], [a17, c17], [a17], [b17, c17], [b17], [c17], []] in its original supplied open-set order, and compact_subset K=[a17, b17]. Qualify the topology and every Hausdorff distinct-point pair. Finite K is compact; the general theorem still needs both ambient Hausdorffness and compactness. Exterior point x=c17 must lie outside K. Give (exterior_point, separation_rows, exterior_neighborhood, cover_union, covers_K, neighborhood_disjoint_from_K, neighborhood_open, certificate).
cover_union may extend beyond K. The exterior neighborhood is a finite intersection and must be open and disjoint from K. No minimum or unique subcover is asserted.
View answer and explanation
(c17, [[a17, [[b17, c17], [a17]]], [b17, [[a17, c17], [b17]]]], [c17], [a17, b17], Yes, Yes, Yes, finite_subcover_and_finite_intersection_separate_x_from_K)
For each y∈K choose disjoint U_y containing x and V_y containing y in original open-list order. The V_y union covers K but may exceed K; intersect the finitely many U_y to obtain an open neighborhood disjoint from K. For empty K the empty intersection is X. No minimum or unique subcover is asserted; the general proof uses compactness to obtain a finite subcover. The exact result, retaining the requested order, is (c17, [[a17, [[b17, c17], [a17]]], [b17, [[a17, c17], [b17]]]], [c17], [a17, b17], Yes, Yes, Yes, finite_subcover_and_finite_intersection_separate_x_from_K).
Reviewed step-by-step derivation
A compact subset of a Hausdorff space is closed · qf-mod43-compactclosed-neighborhood-subcover-certificate · candidate
69. Finite Intersections Separate an Exterior Point from the Whole Compact Subset
Use complete finite labeled carrier=[a18, b18, c18], topology=[[a18, b18, c18], [a18, b18], [a18, c18], [a18], [b18, c18], [b18], [c18], []] in its original supplied open-set order, and compact_subset K=[a18, b18]. Qualify the topology and every Hausdorff distinct-point pair. Finite K is compact; the general theorem still needs both ambient Hausdorffness and compactness. Exterior point x=c18 must lie outside K. Give (exterior_point, separation_rows, exterior_neighborhood, cover_union, covers_K, neighborhood_disjoint_from_K, neighborhood_open, certificate).
cover_union may extend beyond K. The exterior neighborhood is a finite intersection and must be open and disjoint from K. No minimum or unique subcover is asserted.
View answer and explanation
(c18, [[a18, [[b18, c18], [a18]]], [b18, [[a18, c18], [b18]]]], [c18], [a18, b18], Yes, Yes, Yes, finite_subcover_and_finite_intersection_separate_x_from_K)
For each y∈K choose disjoint U_y containing x and V_y containing y in original open-list order. The V_y union covers K but may exceed K; intersect the finitely many U_y to obtain an open neighborhood disjoint from K. For empty K the empty intersection is X. No minimum or unique subcover is asserted; the general proof uses compactness to obtain a finite subcover. The exact result, retaining the requested order, is (c18, [[a18, [[b18, c18], [a18]]], [b18, [[a18, c18], [b18]]]], [c18], [a18, b18], Yes, Yes, Yes, finite_subcover_and_finite_intersection_separate_x_from_K).
Reviewed step-by-step derivation
A compact subset of a Hausdorff space is closed · qf-mod43-compactclosed-neighborhood-subcover-certificate · candidate
70. Finite Intersections Separate an Exterior Point from the Whole Compact Subset
Use complete finite labeled carrier=[a19, b19, c19], topology=[[a19, b19, c19], [a19, b19], [a19, c19], [a19], [b19, c19], [b19], [c19], []] in its original supplied open-set order, and compact_subset K=[a19, b19]. Qualify the topology and every Hausdorff distinct-point pair. Finite K is compact; the general theorem still needs both ambient Hausdorffness and compactness. Exterior point x=c19 must lie outside K. Give (exterior_point, separation_rows, exterior_neighborhood, cover_union, covers_K, neighborhood_disjoint_from_K, neighborhood_open, certificate).
cover_union may extend beyond K. The exterior neighborhood is a finite intersection and must be open and disjoint from K. No minimum or unique subcover is asserted.
View answer and explanation
(c19, [[a19, [[b19, c19], [a19]]], [b19, [[a19, c19], [b19]]]], [c19], [a19, b19], Yes, Yes, Yes, finite_subcover_and_finite_intersection_separate_x_from_K)
For each y∈K choose disjoint U_y containing x and V_y containing y in original open-list order. The V_y union covers K but may exceed K; intersect the finitely many U_y to obtain an open neighborhood disjoint from K. For empty K the empty intersection is X. No minimum or unique subcover is asserted; the general proof uses compactness to obtain a finite subcover. The exact result, retaining the requested order, is (c19, [[a19, [[b19, c19], [a19]]], [b19, [[a19, c19], [b19]]]], [c19], [a19, b19], Yes, Yes, Yes, finite_subcover_and_finite_intersection_separate_x_from_K).
Reviewed step-by-step derivation
A compact subset of a Hausdorff space is closed · qf-mod43-compactclosed-neighborhood-subcover-certificate · candidate
71. Finite Intersections Separate an Exterior Point from the Whole Compact Subset
Use complete finite labeled carrier=[a20, b20, c20], topology=[[a20, b20, c20], [a20, b20], [a20, c20], [a20], [b20, c20], [b20], [c20], []] in its original supplied open-set order, and compact_subset K=[a20, b20]. Qualify the topology and every Hausdorff distinct-point pair. Finite K is compact; the general theorem still needs both ambient Hausdorffness and compactness. Exterior point x=c20 must lie outside K. Give (exterior_point, separation_rows, exterior_neighborhood, cover_union, covers_K, neighborhood_disjoint_from_K, neighborhood_open, certificate).
cover_union may extend beyond K. The exterior neighborhood is a finite intersection and must be open and disjoint from K. No minimum or unique subcover is asserted.
View answer and explanation
(c20, [[a20, [[b20, c20], [a20]]], [b20, [[a20, c20], [b20]]]], [c20], [a20, b20], Yes, Yes, Yes, finite_subcover_and_finite_intersection_separate_x_from_K)
For each y∈K choose disjoint U_y containing x and V_y containing y in original open-list order. The V_y union covers K but may exceed K; intersect the finitely many U_y to obtain an open neighborhood disjoint from K. For empty K the empty intersection is X. No minimum or unique subcover is asserted; the general proof uses compactness to obtain a finite subcover. The exact result, retaining the requested order, is (c20, [[a20, [[b20, c20], [a20]]], [b20, [[a20, c20], [b20]]]], [c20], [a20, b20], Yes, Yes, Yes, finite_subcover_and_finite_intersection_separate_x_from_K).
Reviewed step-by-step derivation
A compact subset of a Hausdorff space is closed · qf-mod43-compactclosed-neighborhood-subcover-certificate · candidate
72. Finite Intersections Separate an Exterior Point from the Whole Compact Subset
Use complete finite labeled carrier=[a21, b21, c21], topology=[[a21, b21, c21], [a21, b21], [a21, c21], [a21], [b21, c21], [b21], [c21], []] in its original supplied open-set order, and compact_subset K=[a21, b21]. Qualify the topology and every Hausdorff distinct-point pair. Finite K is compact; the general theorem still needs both ambient Hausdorffness and compactness. Exterior point x=c21 must lie outside K. Give (exterior_point, separation_rows, exterior_neighborhood, cover_union, covers_K, neighborhood_disjoint_from_K, neighborhood_open, certificate).
cover_union may extend beyond K. The exterior neighborhood is a finite intersection and must be open and disjoint from K. No minimum or unique subcover is asserted.
View answer and explanation
(c21, [[a21, [[b21, c21], [a21]]], [b21, [[a21, c21], [b21]]]], [c21], [a21, b21], Yes, Yes, Yes, finite_subcover_and_finite_intersection_separate_x_from_K)
For each y∈K choose disjoint U_y containing x and V_y containing y in original open-list order. The V_y union covers K but may exceed K; intersect the finitely many U_y to obtain an open neighborhood disjoint from K. For empty K the empty intersection is X. No minimum or unique subcover is asserted; the general proof uses compactness to obtain a finite subcover. The exact result, retaining the requested order, is (c21, [[a21, [[b21, c21], [a21]]], [b21, [[a21, c21], [b21]]]], [c21], [a21, b21], Yes, Yes, Yes, finite_subcover_and_finite_intersection_separate_x_from_K).
Reviewed step-by-step derivation
A compact subset of a Hausdorff space is closed · qf-mod43-compactclosed-neighborhood-subcover-certificate · candidate
73. Finite Intersections Separate an Exterior Point from the Whole Compact Subset
Use complete finite labeled carrier=[a22, b22, c22], topology=[[a22, b22, c22], [a22, b22], [a22, c22], [a22], [b22, c22], [b22], [c22], []] in its original supplied open-set order, and compact_subset K=[a22, b22]. Qualify the topology and every Hausdorff distinct-point pair. Finite K is compact; the general theorem still needs both ambient Hausdorffness and compactness. Exterior point x=c22 must lie outside K. Give (exterior_point, separation_rows, exterior_neighborhood, cover_union, covers_K, neighborhood_disjoint_from_K, neighborhood_open, certificate).
cover_union may extend beyond K. The exterior neighborhood is a finite intersection and must be open and disjoint from K. No minimum or unique subcover is asserted.
View answer and explanation
(c22, [[a22, [[b22, c22], [a22]]], [b22, [[a22, c22], [b22]]]], [c22], [a22, b22], Yes, Yes, Yes, finite_subcover_and_finite_intersection_separate_x_from_K)
For each y∈K choose disjoint U_y containing x and V_y containing y in original open-list order. The V_y union covers K but may exceed K; intersect the finitely many U_y to obtain an open neighborhood disjoint from K. For empty K the empty intersection is X. No minimum or unique subcover is asserted; the general proof uses compactness to obtain a finite subcover. The exact result, retaining the requested order, is (c22, [[a22, [[b22, c22], [a22]]], [b22, [[a22, c22], [b22]]]], [c22], [a22, b22], Yes, Yes, Yes, finite_subcover_and_finite_intersection_separate_x_from_K).
Reviewed step-by-step derivation
A compact subset of a Hausdorff space is closed · qf-mod43-compactclosed-neighborhood-subcover-certificate · candidate
74. Finite Intersections Separate an Exterior Point from the Whole Compact Subset
Use complete finite labeled carrier=[a23, b23, c23], topology=[[a23, b23, c23], [a23, b23], [a23, c23], [a23], [b23, c23], [b23], [c23], []] in its original supplied open-set order, and compact_subset K=[a23, b23]. Qualify the topology and every Hausdorff distinct-point pair. Finite K is compact; the general theorem still needs both ambient Hausdorffness and compactness. Exterior point x=c23 must lie outside K. Give (exterior_point, separation_rows, exterior_neighborhood, cover_union, covers_K, neighborhood_disjoint_from_K, neighborhood_open, certificate).
cover_union may extend beyond K. The exterior neighborhood is a finite intersection and must be open and disjoint from K. No minimum or unique subcover is asserted.
View answer and explanation
(c23, [[a23, [[b23, c23], [a23]]], [b23, [[a23, c23], [b23]]]], [c23], [a23, b23], Yes, Yes, Yes, finite_subcover_and_finite_intersection_separate_x_from_K)
For each y∈K choose disjoint U_y containing x and V_y containing y in original open-list order. The V_y union covers K but may exceed K; intersect the finitely many U_y to obtain an open neighborhood disjoint from K. For empty K the empty intersection is X. No minimum or unique subcover is asserted; the general proof uses compactness to obtain a finite subcover. The exact result, retaining the requested order, is (c23, [[a23, [[b23, c23], [a23]]], [b23, [[a23, c23], [b23]]]], [c23], [a23, b23], Yes, Yes, Yes, finite_subcover_and_finite_intersection_separate_x_from_K).
Reviewed step-by-step derivation
A compact subset of a Hausdorff space is closed · qf-mod43-compactclosed-neighborhood-subcover-certificate · candidate
75. Finite Intersections Separate an Exterior Point from the Whole Compact Subset
Use complete finite labeled carrier=[a24, b24, c24], topology=[[a24, b24, c24], [a24, b24], [a24, c24], [a24], [b24, c24], [b24], [c24], []] in its original supplied open-set order, and compact_subset K=[a24, b24]. Qualify the topology and every Hausdorff distinct-point pair. Finite K is compact; the general theorem still needs both ambient Hausdorffness and compactness. Exterior point x=c24 must lie outside K. Give (exterior_point, separation_rows, exterior_neighborhood, cover_union, covers_K, neighborhood_disjoint_from_K, neighborhood_open, certificate).
cover_union may extend beyond K. The exterior neighborhood is a finite intersection and must be open and disjoint from K. No minimum or unique subcover is asserted.
View answer and explanation
(c24, [[a24, [[b24, c24], [a24]]], [b24, [[a24, c24], [b24]]]], [c24], [a24, b24], Yes, Yes, Yes, finite_subcover_and_finite_intersection_separate_x_from_K)
For each y∈K choose disjoint U_y containing x and V_y containing y in original open-list order. The V_y union covers K but may exceed K; intersect the finitely many U_y to obtain an open neighborhood disjoint from K. For empty K the empty intersection is X. No minimum or unique subcover is asserted; the general proof uses compactness to obtain a finite subcover. The exact result, retaining the requested order, is (c24, [[a24, [[b24, c24], [a24]]], [b24, [[a24, c24], [b24]]]], [c24], [a24, b24], Yes, Yes, Yes, finite_subcover_and_finite_intersection_separate_x_from_K).
Reviewed step-by-step derivation
A compact subset of a Hausdorff space is closed · qf-mod43-compactclosed-neighborhood-subcover-certificate · candidate
76. A Non-Hausdorff Compact Nonclosed Instance Blocks Deleting the Hypothesis
Use complete finite labeled carrier=[o0, u0], topology=[[o0, u0], [u0], []] in its original supplied open-set order, and compact_subset K=[u0]. Qualify the topology and every Hausdorff distinct-point pair. Finite K is compact; the general theorem still needs both ambient Hausdorffness and compactness. Give (points, compact_subset, complement, Hausdorff, compact, closed, boundary).
View answer and explanation
([o0, u0], [u0], [o0], No, Yes, No, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)
Require this actual finite space to be non-Hausdorff and K not closed. K remains compact by finiteness. This is a counterexample to removing the Hausdorff hypothesis, not a claim that every non-Hausdorff compact subset is nonclosed or a converse counterexample. The exact result, retaining the requested order, is ([o0, u0], [u0], [o0], No, Yes, No, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed).
Reviewed step-by-step derivation
A compact subset of a Hausdorff space is closed · qf-mod43-compactclosed-nonhausdorff-boundary · candidate
77. A Non-Hausdorff Compact Nonclosed Instance Blocks Deleting the Hypothesis
Use complete finite labeled carrier=[o1, u1], topology=[[o1, u1], [u1], []] in its original supplied open-set order, and compact_subset K=[u1]. Qualify the topology and every Hausdorff distinct-point pair. Finite K is compact; the general theorem still needs both ambient Hausdorffness and compactness. Give (points, compact_subset, complement, Hausdorff, compact, closed, boundary).
View answer and explanation
([o1, u1], [u1], [o1], No, Yes, No, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)
Require this actual finite space to be non-Hausdorff and K not closed. K remains compact by finiteness. This is a counterexample to removing the Hausdorff hypothesis, not a claim that every non-Hausdorff compact subset is nonclosed or a converse counterexample. The exact result, retaining the requested order, is ([o1, u1], [u1], [o1], No, Yes, No, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed).
Reviewed step-by-step derivation
A compact subset of a Hausdorff space is closed · qf-mod43-compactclosed-nonhausdorff-boundary · candidate
78. A Non-Hausdorff Compact Nonclosed Instance Blocks Deleting the Hypothesis
Use complete finite labeled carrier=[o2, u2], topology=[[o2, u2], [u2], []] in its original supplied open-set order, and compact_subset K=[u2]. Qualify the topology and every Hausdorff distinct-point pair. Finite K is compact; the general theorem still needs both ambient Hausdorffness and compactness. Give (points, compact_subset, complement, Hausdorff, compact, closed, boundary).
View answer and explanation
([o2, u2], [u2], [o2], No, Yes, No, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)
Require this actual finite space to be non-Hausdorff and K not closed. K remains compact by finiteness. This is a counterexample to removing the Hausdorff hypothesis, not a claim that every non-Hausdorff compact subset is nonclosed or a converse counterexample. The exact result, retaining the requested order, is ([o2, u2], [u2], [o2], No, Yes, No, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed).
Reviewed step-by-step derivation
A compact subset of a Hausdorff space is closed · qf-mod43-compactclosed-nonhausdorff-boundary · candidate
79. A Non-Hausdorff Compact Nonclosed Instance Blocks Deleting the Hypothesis
Use complete finite labeled carrier=[o3, u3], topology=[[o3, u3], [u3], []] in its original supplied open-set order, and compact_subset K=[u3]. Qualify the topology and every Hausdorff distinct-point pair. Finite K is compact; the general theorem still needs both ambient Hausdorffness and compactness. Give (points, compact_subset, complement, Hausdorff, compact, closed, boundary).
View answer and explanation
([o3, u3], [u3], [o3], No, Yes, No, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)
Require this actual finite space to be non-Hausdorff and K not closed. K remains compact by finiteness. This is a counterexample to removing the Hausdorff hypothesis, not a claim that every non-Hausdorff compact subset is nonclosed or a converse counterexample. The exact result, retaining the requested order, is ([o3, u3], [u3], [o3], No, Yes, No, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed).
Reviewed step-by-step derivation
A compact subset of a Hausdorff space is closed · qf-mod43-compactclosed-nonhausdorff-boundary · candidate
80. A Non-Hausdorff Compact Nonclosed Instance Blocks Deleting the Hypothesis
Use complete finite labeled carrier=[o4, u4], topology=[[o4, u4], [u4], []] in its original supplied open-set order, and compact_subset K=[u4]. Qualify the topology and every Hausdorff distinct-point pair. Finite K is compact; the general theorem still needs both ambient Hausdorffness and compactness. Give (points, compact_subset, complement, Hausdorff, compact, closed, boundary).
View answer and explanation
([o4, u4], [u4], [o4], No, Yes, No, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)
Require this actual finite space to be non-Hausdorff and K not closed. K remains compact by finiteness. This is a counterexample to removing the Hausdorff hypothesis, not a claim that every non-Hausdorff compact subset is nonclosed or a converse counterexample. The exact result, retaining the requested order, is ([o4, u4], [u4], [o4], No, Yes, No, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed).
Reviewed step-by-step derivation
A compact subset of a Hausdorff space is closed · qf-mod43-compactclosed-nonhausdorff-boundary · candidate
81. A Non-Hausdorff Compact Nonclosed Instance Blocks Deleting the Hypothesis
Use complete finite labeled carrier=[o5, u5], topology=[[o5, u5], [u5], []] in its original supplied open-set order, and compact_subset K=[u5]. Qualify the topology and every Hausdorff distinct-point pair. Finite K is compact; the general theorem still needs both ambient Hausdorffness and compactness. Give (points, compact_subset, complement, Hausdorff, compact, closed, boundary).
View answer and explanation
([o5, u5], [u5], [o5], No, Yes, No, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)
Require this actual finite space to be non-Hausdorff and K not closed. K remains compact by finiteness. This is a counterexample to removing the Hausdorff hypothesis, not a claim that every non-Hausdorff compact subset is nonclosed or a converse counterexample. The exact result, retaining the requested order, is ([o5, u5], [u5], [o5], No, Yes, No, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed).
Reviewed step-by-step derivation
A compact subset of a Hausdorff space is closed · qf-mod43-compactclosed-nonhausdorff-boundary · candidate
82. A Non-Hausdorff Compact Nonclosed Instance Blocks Deleting the Hypothesis
Use complete finite labeled carrier=[o6, u6], topology=[[o6, u6], [u6], []] in its original supplied open-set order, and compact_subset K=[u6]. Qualify the topology and every Hausdorff distinct-point pair. Finite K is compact; the general theorem still needs both ambient Hausdorffness and compactness. Give (points, compact_subset, complement, Hausdorff, compact, closed, boundary).
View answer and explanation
([o6, u6], [u6], [o6], No, Yes, No, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)
Require this actual finite space to be non-Hausdorff and K not closed. K remains compact by finiteness. This is a counterexample to removing the Hausdorff hypothesis, not a claim that every non-Hausdorff compact subset is nonclosed or a converse counterexample. The exact result, retaining the requested order, is ([o6, u6], [u6], [o6], No, Yes, No, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed).
Reviewed step-by-step derivation
A compact subset of a Hausdorff space is closed · qf-mod43-compactclosed-nonhausdorff-boundary · candidate
83. A Non-Hausdorff Compact Nonclosed Instance Blocks Deleting the Hypothesis
Use complete finite labeled carrier=[o7, u7], topology=[[o7, u7], [u7], []] in its original supplied open-set order, and compact_subset K=[u7]. Qualify the topology and every Hausdorff distinct-point pair. Finite K is compact; the general theorem still needs both ambient Hausdorffness and compactness. Give (points, compact_subset, complement, Hausdorff, compact, closed, boundary).
View answer and explanation
([o7, u7], [u7], [o7], No, Yes, No, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)
Require this actual finite space to be non-Hausdorff and K not closed. K remains compact by finiteness. This is a counterexample to removing the Hausdorff hypothesis, not a claim that every non-Hausdorff compact subset is nonclosed or a converse counterexample. The exact result, retaining the requested order, is ([o7, u7], [u7], [o7], No, Yes, No, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed).
Reviewed step-by-step derivation
A compact subset of a Hausdorff space is closed · qf-mod43-compactclosed-nonhausdorff-boundary · candidate
84. A Non-Hausdorff Compact Nonclosed Instance Blocks Deleting the Hypothesis
Use complete finite labeled carrier=[o8, u8], topology=[[o8, u8], [u8], []] in its original supplied open-set order, and compact_subset K=[u8]. Qualify the topology and every Hausdorff distinct-point pair. Finite K is compact; the general theorem still needs both ambient Hausdorffness and compactness. Give (points, compact_subset, complement, Hausdorff, compact, closed, boundary).
View answer and explanation
([o8, u8], [u8], [o8], No, Yes, No, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)
Require this actual finite space to be non-Hausdorff and K not closed. K remains compact by finiteness. This is a counterexample to removing the Hausdorff hypothesis, not a claim that every non-Hausdorff compact subset is nonclosed or a converse counterexample. The exact result, retaining the requested order, is ([o8, u8], [u8], [o8], No, Yes, No, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed).
Reviewed step-by-step derivation
A compact subset of a Hausdorff space is closed · qf-mod43-compactclosed-nonhausdorff-boundary · candidate
85. A Non-Hausdorff Compact Nonclosed Instance Blocks Deleting the Hypothesis
Use complete finite labeled carrier=[o9, u9], topology=[[o9, u9], [u9], []] in its original supplied open-set order, and compact_subset K=[u9]. Qualify the topology and every Hausdorff distinct-point pair. Finite K is compact; the general theorem still needs both ambient Hausdorffness and compactness. Give (points, compact_subset, complement, Hausdorff, compact, closed, boundary).
View answer and explanation
([o9, u9], [u9], [o9], No, Yes, No, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)
Require this actual finite space to be non-Hausdorff and K not closed. K remains compact by finiteness. This is a counterexample to removing the Hausdorff hypothesis, not a claim that every non-Hausdorff compact subset is nonclosed or a converse counterexample. The exact result, retaining the requested order, is ([o9, u9], [u9], [o9], No, Yes, No, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed).
Reviewed step-by-step derivation
A compact subset of a Hausdorff space is closed · qf-mod43-compactclosed-nonhausdorff-boundary · candidate
86. A Non-Hausdorff Compact Nonclosed Instance Blocks Deleting the Hypothesis
Use complete finite labeled carrier=[o10, u10], topology=[[o10, u10], [u10], []] in its original supplied open-set order, and compact_subset K=[u10]. Qualify the topology and every Hausdorff distinct-point pair. Finite K is compact; the general theorem still needs both ambient Hausdorffness and compactness. Give (points, compact_subset, complement, Hausdorff, compact, closed, boundary).
View answer and explanation
([o10, u10], [u10], [o10], No, Yes, No, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)
Require this actual finite space to be non-Hausdorff and K not closed. K remains compact by finiteness. This is a counterexample to removing the Hausdorff hypothesis, not a claim that every non-Hausdorff compact subset is nonclosed or a converse counterexample. The exact result, retaining the requested order, is ([o10, u10], [u10], [o10], No, Yes, No, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed).
Reviewed step-by-step derivation
A compact subset of a Hausdorff space is closed · qf-mod43-compactclosed-nonhausdorff-boundary · candidate
87. A Non-Hausdorff Compact Nonclosed Instance Blocks Deleting the Hypothesis
Use complete finite labeled carrier=[o11, u11], topology=[[o11, u11], [u11], []] in its original supplied open-set order, and compact_subset K=[u11]. Qualify the topology and every Hausdorff distinct-point pair. Finite K is compact; the general theorem still needs both ambient Hausdorffness and compactness. Give (points, compact_subset, complement, Hausdorff, compact, closed, boundary).
View answer and explanation
([o11, u11], [u11], [o11], No, Yes, No, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)
Require this actual finite space to be non-Hausdorff and K not closed. K remains compact by finiteness. This is a counterexample to removing the Hausdorff hypothesis, not a claim that every non-Hausdorff compact subset is nonclosed or a converse counterexample. The exact result, retaining the requested order, is ([o11, u11], [u11], [o11], No, Yes, No, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed).
Reviewed step-by-step derivation
A compact subset of a Hausdorff space is closed · qf-mod43-compactclosed-nonhausdorff-boundary · candidate
88. A Non-Hausdorff Compact Nonclosed Instance Blocks Deleting the Hypothesis
Use complete finite labeled carrier=[o12, u12], topology=[[o12, u12], [u12], []] in its original supplied open-set order, and compact_subset K=[u12]. Qualify the topology and every Hausdorff distinct-point pair. Finite K is compact; the general theorem still needs both ambient Hausdorffness and compactness. Give (points, compact_subset, complement, Hausdorff, compact, closed, boundary).
View answer and explanation
([o12, u12], [u12], [o12], No, Yes, No, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)
Require this actual finite space to be non-Hausdorff and K not closed. K remains compact by finiteness. This is a counterexample to removing the Hausdorff hypothesis, not a claim that every non-Hausdorff compact subset is nonclosed or a converse counterexample. The exact result, retaining the requested order, is ([o12, u12], [u12], [o12], No, Yes, No, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed).
Reviewed step-by-step derivation
A compact subset of a Hausdorff space is closed · qf-mod43-compactclosed-nonhausdorff-boundary · candidate
89. A Non-Hausdorff Compact Nonclosed Instance Blocks Deleting the Hypothesis
Use complete finite labeled carrier=[o13, u13], topology=[[o13, u13], [u13], []] in its original supplied open-set order, and compact_subset K=[u13]. Qualify the topology and every Hausdorff distinct-point pair. Finite K is compact; the general theorem still needs both ambient Hausdorffness and compactness. Give (points, compact_subset, complement, Hausdorff, compact, closed, boundary).
View answer and explanation
([o13, u13], [u13], [o13], No, Yes, No, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)
Require this actual finite space to be non-Hausdorff and K not closed. K remains compact by finiteness. This is a counterexample to removing the Hausdorff hypothesis, not a claim that every non-Hausdorff compact subset is nonclosed or a converse counterexample. The exact result, retaining the requested order, is ([o13, u13], [u13], [o13], No, Yes, No, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed).
Reviewed step-by-step derivation
A compact subset of a Hausdorff space is closed · qf-mod43-compactclosed-nonhausdorff-boundary · candidate
90. A Non-Hausdorff Compact Nonclosed Instance Blocks Deleting the Hypothesis
Use complete finite labeled carrier=[o14, u14], topology=[[o14, u14], [u14], []] in its original supplied open-set order, and compact_subset K=[u14]. Qualify the topology and every Hausdorff distinct-point pair. Finite K is compact; the general theorem still needs both ambient Hausdorffness and compactness. Give (points, compact_subset, complement, Hausdorff, compact, closed, boundary).
View answer and explanation
([o14, u14], [u14], [o14], No, Yes, No, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)
Require this actual finite space to be non-Hausdorff and K not closed. K remains compact by finiteness. This is a counterexample to removing the Hausdorff hypothesis, not a claim that every non-Hausdorff compact subset is nonclosed or a converse counterexample. The exact result, retaining the requested order, is ([o14, u14], [u14], [o14], No, Yes, No, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed).
Reviewed step-by-step derivation
A compact subset of a Hausdorff space is closed · qf-mod43-compactclosed-nonhausdorff-boundary · candidate
91. A Non-Hausdorff Compact Nonclosed Instance Blocks Deleting the Hypothesis
Use complete finite labeled carrier=[o15, u15], topology=[[o15, u15], [u15], []] in its original supplied open-set order, and compact_subset K=[u15]. Qualify the topology and every Hausdorff distinct-point pair. Finite K is compact; the general theorem still needs both ambient Hausdorffness and compactness. Give (points, compact_subset, complement, Hausdorff, compact, closed, boundary).
View answer and explanation
([o15, u15], [u15], [o15], No, Yes, No, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)
Require this actual finite space to be non-Hausdorff and K not closed. K remains compact by finiteness. This is a counterexample to removing the Hausdorff hypothesis, not a claim that every non-Hausdorff compact subset is nonclosed or a converse counterexample. The exact result, retaining the requested order, is ([o15, u15], [u15], [o15], No, Yes, No, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed).
Reviewed step-by-step derivation
A compact subset of a Hausdorff space is closed · qf-mod43-compactclosed-nonhausdorff-boundary · candidate
92. A Non-Hausdorff Compact Nonclosed Instance Blocks Deleting the Hypothesis
Use complete finite labeled carrier=[o16, u16], topology=[[o16, u16], [u16], []] in its original supplied open-set order, and compact_subset K=[u16]. Qualify the topology and every Hausdorff distinct-point pair. Finite K is compact; the general theorem still needs both ambient Hausdorffness and compactness. Give (points, compact_subset, complement, Hausdorff, compact, closed, boundary).
View answer and explanation
([o16, u16], [u16], [o16], No, Yes, No, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)
Require this actual finite space to be non-Hausdorff and K not closed. K remains compact by finiteness. This is a counterexample to removing the Hausdorff hypothesis, not a claim that every non-Hausdorff compact subset is nonclosed or a converse counterexample. The exact result, retaining the requested order, is ([o16, u16], [u16], [o16], No, Yes, No, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed).
Reviewed step-by-step derivation
A compact subset of a Hausdorff space is closed · qf-mod43-compactclosed-nonhausdorff-boundary · candidate
93. A Non-Hausdorff Compact Nonclosed Instance Blocks Deleting the Hypothesis
Use complete finite labeled carrier=[o17, u17], topology=[[o17, u17], [u17], []] in its original supplied open-set order, and compact_subset K=[u17]. Qualify the topology and every Hausdorff distinct-point pair. Finite K is compact; the general theorem still needs both ambient Hausdorffness and compactness. Give (points, compact_subset, complement, Hausdorff, compact, closed, boundary).
View answer and explanation
([o17, u17], [u17], [o17], No, Yes, No, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)
Require this actual finite space to be non-Hausdorff and K not closed. K remains compact by finiteness. This is a counterexample to removing the Hausdorff hypothesis, not a claim that every non-Hausdorff compact subset is nonclosed or a converse counterexample. The exact result, retaining the requested order, is ([o17, u17], [u17], [o17], No, Yes, No, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed).
Reviewed step-by-step derivation
A compact subset of a Hausdorff space is closed · qf-mod43-compactclosed-nonhausdorff-boundary · candidate
94. A Non-Hausdorff Compact Nonclosed Instance Blocks Deleting the Hypothesis
Use complete finite labeled carrier=[o18, u18], topology=[[o18, u18], [u18], []] in its original supplied open-set order, and compact_subset K=[u18]. Qualify the topology and every Hausdorff distinct-point pair. Finite K is compact; the general theorem still needs both ambient Hausdorffness and compactness. Give (points, compact_subset, complement, Hausdorff, compact, closed, boundary).
View answer and explanation
([o18, u18], [u18], [o18], No, Yes, No, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)
Require this actual finite space to be non-Hausdorff and K not closed. K remains compact by finiteness. This is a counterexample to removing the Hausdorff hypothesis, not a claim that every non-Hausdorff compact subset is nonclosed or a converse counterexample. The exact result, retaining the requested order, is ([o18, u18], [u18], [o18], No, Yes, No, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed).
Reviewed step-by-step derivation
A compact subset of a Hausdorff space is closed · qf-mod43-compactclosed-nonhausdorff-boundary · candidate
95. A Non-Hausdorff Compact Nonclosed Instance Blocks Deleting the Hypothesis
Use complete finite labeled carrier=[o19, u19], topology=[[o19, u19], [u19], []] in its original supplied open-set order, and compact_subset K=[u19]. Qualify the topology and every Hausdorff distinct-point pair. Finite K is compact; the general theorem still needs both ambient Hausdorffness and compactness. Give (points, compact_subset, complement, Hausdorff, compact, closed, boundary).
View answer and explanation
([o19, u19], [u19], [o19], No, Yes, No, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)
Require this actual finite space to be non-Hausdorff and K not closed. K remains compact by finiteness. This is a counterexample to removing the Hausdorff hypothesis, not a claim that every non-Hausdorff compact subset is nonclosed or a converse counterexample. The exact result, retaining the requested order, is ([o19, u19], [u19], [o19], No, Yes, No, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed).
Reviewed step-by-step derivation
A compact subset of a Hausdorff space is closed · qf-mod43-compactclosed-nonhausdorff-boundary · candidate
96. A Non-Hausdorff Compact Nonclosed Instance Blocks Deleting the Hypothesis
Use complete finite labeled carrier=[o20, u20], topology=[[o20, u20], [u20], []] in its original supplied open-set order, and compact_subset K=[u20]. Qualify the topology and every Hausdorff distinct-point pair. Finite K is compact; the general theorem still needs both ambient Hausdorffness and compactness. Give (points, compact_subset, complement, Hausdorff, compact, closed, boundary).
View answer and explanation
([o20, u20], [u20], [o20], No, Yes, No, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)
Require this actual finite space to be non-Hausdorff and K not closed. K remains compact by finiteness. This is a counterexample to removing the Hausdorff hypothesis, not a claim that every non-Hausdorff compact subset is nonclosed or a converse counterexample. The exact result, retaining the requested order, is ([o20, u20], [u20], [o20], No, Yes, No, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed).
Reviewed step-by-step derivation
A compact subset of a Hausdorff space is closed · qf-mod43-compactclosed-nonhausdorff-boundary · candidate
97. A Non-Hausdorff Compact Nonclosed Instance Blocks Deleting the Hypothesis
Use complete finite labeled carrier=[o21, u21], topology=[[o21, u21], [u21], []] in its original supplied open-set order, and compact_subset K=[u21]. Qualify the topology and every Hausdorff distinct-point pair. Finite K is compact; the general theorem still needs both ambient Hausdorffness and compactness. Give (points, compact_subset, complement, Hausdorff, compact, closed, boundary).
View answer and explanation
([o21, u21], [u21], [o21], No, Yes, No, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)
Require this actual finite space to be non-Hausdorff and K not closed. K remains compact by finiteness. This is a counterexample to removing the Hausdorff hypothesis, not a claim that every non-Hausdorff compact subset is nonclosed or a converse counterexample. The exact result, retaining the requested order, is ([o21, u21], [u21], [o21], No, Yes, No, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed).
Reviewed step-by-step derivation
A compact subset of a Hausdorff space is closed · qf-mod43-compactclosed-nonhausdorff-boundary · candidate
98. A Non-Hausdorff Compact Nonclosed Instance Blocks Deleting the Hypothesis
Use complete finite labeled carrier=[o22, u22], topology=[[o22, u22], [u22], []] in its original supplied open-set order, and compact_subset K=[u22]. Qualify the topology and every Hausdorff distinct-point pair. Finite K is compact; the general theorem still needs both ambient Hausdorffness and compactness. Give (points, compact_subset, complement, Hausdorff, compact, closed, boundary).
View answer and explanation
([o22, u22], [u22], [o22], No, Yes, No, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)
Require this actual finite space to be non-Hausdorff and K not closed. K remains compact by finiteness. This is a counterexample to removing the Hausdorff hypothesis, not a claim that every non-Hausdorff compact subset is nonclosed or a converse counterexample. The exact result, retaining the requested order, is ([o22, u22], [u22], [o22], No, Yes, No, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed).
Reviewed step-by-step derivation
A compact subset of a Hausdorff space is closed · qf-mod43-compactclosed-nonhausdorff-boundary · candidate
99. A Non-Hausdorff Compact Nonclosed Instance Blocks Deleting the Hypothesis
Use complete finite labeled carrier=[o23, u23], topology=[[o23, u23], [u23], []] in its original supplied open-set order, and compact_subset K=[u23]. Qualify the topology and every Hausdorff distinct-point pair. Finite K is compact; the general theorem still needs both ambient Hausdorffness and compactness. Give (points, compact_subset, complement, Hausdorff, compact, closed, boundary).
View answer and explanation
([o23, u23], [u23], [o23], No, Yes, No, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)
Require this actual finite space to be non-Hausdorff and K not closed. K remains compact by finiteness. This is a counterexample to removing the Hausdorff hypothesis, not a claim that every non-Hausdorff compact subset is nonclosed or a converse counterexample. The exact result, retaining the requested order, is ([o23, u23], [u23], [o23], No, Yes, No, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed).
Reviewed step-by-step derivation
A compact subset of a Hausdorff space is closed · qf-mod43-compactclosed-nonhausdorff-boundary · candidate
100. A Non-Hausdorff Compact Nonclosed Instance Blocks Deleting the Hypothesis
Use complete finite labeled carrier=[o24, u24], topology=[[o24, u24], [u24], []] in its original supplied open-set order, and compact_subset K=[u24]. Qualify the topology and every Hausdorff distinct-point pair. Finite K is compact; the general theorem still needs both ambient Hausdorffness and compactness. Give (points, compact_subset, complement, Hausdorff, compact, closed, boundary).
View answer and explanation
([o24, u24], [u24], [o24], No, Yes, No, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)
Require this actual finite space to be non-Hausdorff and K not closed. K remains compact by finiteness. This is a counterexample to removing the Hausdorff hypothesis, not a claim that every non-Hausdorff compact subset is nonclosed or a converse counterexample. The exact result, retaining the requested order, is ([o24, u24], [u24], [o24], No, Yes, No, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed).
Reviewed step-by-step derivation
A compact subset of a Hausdorff space is closed · qf-mod43-compactclosed-nonhausdorff-boundary · candidate
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