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Hausdorff空間中的緊緻集為閉集基礎 · 脈衝 R03

本批已通過 Schema、去重、來源快照與獨立答案重算；仍未標記為人工審定。

本題庫已有對應的中英逐步詳解。原題的 candidate 與英文 ai_candidate 狀態保留；AI 覆核、有限計算與 Lean 形式證明分開標示。

batch-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-foundations · 2026-10-14 · 100 題

欄序為來源綁定的呈現補充；原始 JSONL 與數學答案不變。

欄序與來源綁定資料

1. Hausdorff空間中的緊緻集為閉集

ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-hypotheses-qualification；資料={"points":["a0","b0","c0"],"topology":[["a0","b0","c0"],["a0","b0"],["a0","c0"],["a0"],["b0","c0"],["b0"],["c0"],[]],"compact_subset":["a0","b0"]}。請依 [points, compact_subset, pair_separation_rows, Hausdorff, compact, certificate] 欄序回答。

查看答案與解釋

([a0, b0, c0], [a0, b0], [[a0, b0, [[a0, c0], [b0]]], [a0, c0, [[a0, b0], [c0]]], [b0, c0, [[a0, b0], [c0]]]], 是, 是, complete_finite_topology_Hausdorff_and_finite_subset_compact)

Wave6第六層核對=([a0, b0, c0], [a0, b0], [[a0, b0, [[a0, c0], [b0]]], [a0, c0, [[a0, b0], [c0]]], [b0, c0, [[a0, b0], [c0]]]], 是, 是, complete_finite_topology_Hausdorff_and_finite_subset_compact)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。

本題逐步詳解（AI 二次覆核）

Hausdorff空間中的緊緻集為閉集 · qf-mod43-compactclosed-hypotheses-qualification · candidate

2. Hausdorff空間中的緊緻集為閉集

ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-hypotheses-qualification；資料={"points":["a1","b1","c1"],"topology":[["a1","b1","c1"],["a1","b1"],["a1","c1"],["a1"],["b1","c1"],["b1"],["c1"],[]],"compact_subset":["a1","b1"]}。請依 [points, compact_subset, pair_separation_rows, Hausdorff, compact, certificate] 欄序回答。

查看答案與解釋

([a1, b1, c1], [a1, b1], [[a1, b1, [[a1, c1], [b1]]], [a1, c1, [[a1, b1], [c1]]], [b1, c1, [[a1, b1], [c1]]]], 是, 是, complete_finite_topology_Hausdorff_and_finite_subset_compact)

Wave6第六層核對=([a1, b1, c1], [a1, b1], [[a1, b1, [[a1, c1], [b1]]], [a1, c1, [[a1, b1], [c1]]], [b1, c1, [[a1, b1], [c1]]]], 是, 是, complete_finite_topology_Hausdorff_and_finite_subset_compact)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。

本題逐步詳解（AI 二次覆核）

Hausdorff空間中的緊緻集為閉集 · qf-mod43-compactclosed-hypotheses-qualification · candidate

3. Hausdorff空間中的緊緻集為閉集

ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-hypotheses-qualification；資料={"points":["a2","b2","c2"],"topology":[["a2","b2","c2"],["a2","b2"],["a2","c2"],["a2"],["b2","c2"],["b2"],["c2"],[]],"compact_subset":["a2","b2"]}。請依 [points, compact_subset, pair_separation_rows, Hausdorff, compact, certificate] 欄序回答。

查看答案與解釋

([a2, b2, c2], [a2, b2], [[a2, b2, [[a2, c2], [b2]]], [a2, c2, [[a2, b2], [c2]]], [b2, c2, [[a2, b2], [c2]]]], 是, 是, complete_finite_topology_Hausdorff_and_finite_subset_compact)

Wave6第六層核對=([a2, b2, c2], [a2, b2], [[a2, b2, [[a2, c2], [b2]]], [a2, c2, [[a2, b2], [c2]]], [b2, c2, [[a2, b2], [c2]]]], 是, 是, complete_finite_topology_Hausdorff_and_finite_subset_compact)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。

本題逐步詳解（AI 二次覆核）

Hausdorff空間中的緊緻集為閉集 · qf-mod43-compactclosed-hypotheses-qualification · candidate

4. Hausdorff空間中的緊緻集為閉集

ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-hypotheses-qualification；資料={"points":["a3","b3","c3"],"topology":[["a3","b3","c3"],["a3","b3"],["a3","c3"],["a3"],["b3","c3"],["b3"],["c3"],[]],"compact_subset":["a3","b3"]}。請依 [points, compact_subset, pair_separation_rows, Hausdorff, compact, certificate] 欄序回答。

查看答案與解釋

([a3, b3, c3], [a3, b3], [[a3, b3, [[a3, c3], [b3]]], [a3, c3, [[a3, b3], [c3]]], [b3, c3, [[a3, b3], [c3]]]], 是, 是, complete_finite_topology_Hausdorff_and_finite_subset_compact)

Wave6第六層核對=([a3, b3, c3], [a3, b3], [[a3, b3, [[a3, c3], [b3]]], [a3, c3, [[a3, b3], [c3]]], [b3, c3, [[a3, b3], [c3]]]], 是, 是, complete_finite_topology_Hausdorff_and_finite_subset_compact)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。

本題逐步詳解（AI 二次覆核）

Hausdorff空間中的緊緻集為閉集 · qf-mod43-compactclosed-hypotheses-qualification · candidate

5. Hausdorff空間中的緊緻集為閉集

ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-hypotheses-qualification；資料={"points":["a4","b4","c4"],"topology":[["a4","b4","c4"],["a4","b4"],["a4","c4"],["a4"],["b4","c4"],["b4"],["c4"],[]],"compact_subset":["a4","b4"]}。請依 [points, compact_subset, pair_separation_rows, Hausdorff, compact, certificate] 欄序回答。

查看答案與解釋

([a4, b4, c4], [a4, b4], [[a4, b4, [[a4, c4], [b4]]], [a4, c4, [[a4, b4], [c4]]], [b4, c4, [[a4, b4], [c4]]]], 是, 是, complete_finite_topology_Hausdorff_and_finite_subset_compact)

Wave6第六層核對=([a4, b4, c4], [a4, b4], [[a4, b4, [[a4, c4], [b4]]], [a4, c4, [[a4, b4], [c4]]], [b4, c4, [[a4, b4], [c4]]]], 是, 是, complete_finite_topology_Hausdorff_and_finite_subset_compact)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。

本題逐步詳解（AI 二次覆核）

Hausdorff空間中的緊緻集為閉集 · qf-mod43-compactclosed-hypotheses-qualification · candidate

6. Hausdorff空間中的緊緻集為閉集

ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-hypotheses-qualification；資料={"points":["a5","b5","c5"],"topology":[["a5","b5","c5"],["a5","b5"],["a5","c5"],["a5"],["b5","c5"],["b5"],["c5"],[]],"compact_subset":["a5","b5"]}。請依 [points, compact_subset, pair_separation_rows, Hausdorff, compact, certificate] 欄序回答。

查看答案與解釋

([a5, b5, c5], [a5, b5], [[a5, b5, [[a5, c5], [b5]]], [a5, c5, [[a5, b5], [c5]]], [b5, c5, [[a5, b5], [c5]]]], 是, 是, complete_finite_topology_Hausdorff_and_finite_subset_compact)

Wave6第六層核對=([a5, b5, c5], [a5, b5], [[a5, b5, [[a5, c5], [b5]]], [a5, c5, [[a5, b5], [c5]]], [b5, c5, [[a5, b5], [c5]]]], 是, 是, complete_finite_topology_Hausdorff_and_finite_subset_compact)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。

本題逐步詳解（AI 二次覆核）

Hausdorff空間中的緊緻集為閉集 · qf-mod43-compactclosed-hypotheses-qualification · candidate

7. Hausdorff空間中的緊緻集為閉集

ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-hypotheses-qualification；資料={"points":["a6","b6","c6"],"topology":[["a6","b6","c6"],["a6","b6"],["a6","c6"],["a6"],["b6","c6"],["b6"],["c6"],[]],"compact_subset":["a6","b6"]}。請依 [points, compact_subset, pair_separation_rows, Hausdorff, compact, certificate] 欄序回答。

查看答案與解釋

([a6, b6, c6], [a6, b6], [[a6, b6, [[a6, c6], [b6]]], [a6, c6, [[a6, b6], [c6]]], [b6, c6, [[a6, b6], [c6]]]], 是, 是, complete_finite_topology_Hausdorff_and_finite_subset_compact)

Wave6第六層核對=([a6, b6, c6], [a6, b6], [[a6, b6, [[a6, c6], [b6]]], [a6, c6, [[a6, b6], [c6]]], [b6, c6, [[a6, b6], [c6]]]], 是, 是, complete_finite_topology_Hausdorff_and_finite_subset_compact)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。

本題逐步詳解（AI 二次覆核）

Hausdorff空間中的緊緻集為閉集 · qf-mod43-compactclosed-hypotheses-qualification · candidate

8. Hausdorff空間中的緊緻集為閉集

ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-hypotheses-qualification；資料={"points":["a7","b7","c7"],"topology":[["a7","b7","c7"],["a7","b7"],["a7","c7"],["a7"],["b7","c7"],["b7"],["c7"],[]],"compact_subset":["a7","b7"]}。請依 [points, compact_subset, pair_separation_rows, Hausdorff, compact, certificate] 欄序回答。

查看答案與解釋

([a7, b7, c7], [a7, b7], [[a7, b7, [[a7, c7], [b7]]], [a7, c7, [[a7, b7], [c7]]], [b7, c7, [[a7, b7], [c7]]]], 是, 是, complete_finite_topology_Hausdorff_and_finite_subset_compact)

Wave6第六層核對=([a7, b7, c7], [a7, b7], [[a7, b7, [[a7, c7], [b7]]], [a7, c7, [[a7, b7], [c7]]], [b7, c7, [[a7, b7], [c7]]]], 是, 是, complete_finite_topology_Hausdorff_and_finite_subset_compact)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。

本題逐步詳解（AI 二次覆核）

Hausdorff空間中的緊緻集為閉集 · qf-mod43-compactclosed-hypotheses-qualification · candidate

9. Hausdorff空間中的緊緻集為閉集

ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-hypotheses-qualification；資料={"points":["a8","b8","c8"],"topology":[["a8","b8","c8"],["a8","b8"],["a8","c8"],["a8"],["b8","c8"],["b8"],["c8"],[]],"compact_subset":["a8","b8"]}。請依 [points, compact_subset, pair_separation_rows, Hausdorff, compact, certificate] 欄序回答。

查看答案與解釋

([a8, b8, c8], [a8, b8], [[a8, b8, [[a8, c8], [b8]]], [a8, c8, [[a8, b8], [c8]]], [b8, c8, [[a8, b8], [c8]]]], 是, 是, complete_finite_topology_Hausdorff_and_finite_subset_compact)

Wave6第六層核對=([a8, b8, c8], [a8, b8], [[a8, b8, [[a8, c8], [b8]]], [a8, c8, [[a8, b8], [c8]]], [b8, c8, [[a8, b8], [c8]]]], 是, 是, complete_finite_topology_Hausdorff_and_finite_subset_compact)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。

本題逐步詳解（AI 二次覆核）

Hausdorff空間中的緊緻集為閉集 · qf-mod43-compactclosed-hypotheses-qualification · candidate

10. Hausdorff空間中的緊緻集為閉集

ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-hypotheses-qualification；資料={"points":["a9","b9","c9"],"topology":[["a9","b9","c9"],["a9","b9"],["a9","c9"],["a9"],["b9","c9"],["b9"],["c9"],[]],"compact_subset":["a9","b9"]}。請依 [points, compact_subset, pair_separation_rows, Hausdorff, compact, certificate] 欄序回答。

查看答案與解釋

([a9, b9, c9], [a9, b9], [[a9, b9, [[a9, c9], [b9]]], [a9, c9, [[a9, b9], [c9]]], [b9, c9, [[a9, b9], [c9]]]], 是, 是, complete_finite_topology_Hausdorff_and_finite_subset_compact)

Wave6第六層核對=([a9, b9, c9], [a9, b9], [[a9, b9, [[a9, c9], [b9]]], [a9, c9, [[a9, b9], [c9]]], [b9, c9, [[a9, b9], [c9]]]], 是, 是, complete_finite_topology_Hausdorff_and_finite_subset_compact)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。

本題逐步詳解（AI 二次覆核）

Hausdorff空間中的緊緻集為閉集 · qf-mod43-compactclosed-hypotheses-qualification · candidate

11. Hausdorff空間中的緊緻集為閉集

ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-hypotheses-qualification；資料={"points":["a10","b10","c10"],"topology":[["a10","b10","c10"],["a10","b10"],["a10","c10"],["a10"],["b10","c10"],["b10"],["c10"],[]],"compact_subset":["a10","b10"]}。請依 [points, compact_subset, pair_separation_rows, Hausdorff, compact, certificate] 欄序回答。

查看答案與解釋

([a10, b10, c10], [a10, b10], [[a10, b10, [[a10, c10], [b10]]], [a10, c10, [[a10, b10], [c10]]], [b10, c10, [[a10, b10], [c10]]]], 是, 是, complete_finite_topology_Hausdorff_and_finite_subset_compact)

Wave6第六層核對=([a10, b10, c10], [a10, b10], [[a10, b10, [[a10, c10], [b10]]], [a10, c10, [[a10, b10], [c10]]], [b10, c10, [[a10, b10], [c10]]]], 是, 是, complete_finite_topology_Hausdorff_and_finite_subset_compact)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。

本題逐步詳解（AI 二次覆核）

Hausdorff空間中的緊緻集為閉集 · qf-mod43-compactclosed-hypotheses-qualification · candidate

12. Hausdorff空間中的緊緻集為閉集

ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-hypotheses-qualification；資料={"points":["a11","b11","c11"],"topology":[["a11","b11","c11"],["a11","b11"],["a11","c11"],["a11"],["b11","c11"],["b11"],["c11"],[]],"compact_subset":["a11","b11"]}。請依 [points, compact_subset, pair_separation_rows, Hausdorff, compact, certificate] 欄序回答。

查看答案與解釋

([a11, b11, c11], [a11, b11], [[a11, b11, [[a11, c11], [b11]]], [a11, c11, [[a11, b11], [c11]]], [b11, c11, [[a11, b11], [c11]]]], 是, 是, complete_finite_topology_Hausdorff_and_finite_subset_compact)

Wave6第六層核對=([a11, b11, c11], [a11, b11], [[a11, b11, [[a11, c11], [b11]]], [a11, c11, [[a11, b11], [c11]]], [b11, c11, [[a11, b11], [c11]]]], 是, 是, complete_finite_topology_Hausdorff_and_finite_subset_compact)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。

本題逐步詳解（AI 二次覆核）

Hausdorff空間中的緊緻集為閉集 · qf-mod43-compactclosed-hypotheses-qualification · candidate

13. Hausdorff空間中的緊緻集為閉集

ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-hypotheses-qualification；資料={"points":["a12","b12","c12"],"topology":[["a12","b12","c12"],["a12","b12"],["a12","c12"],["a12"],["b12","c12"],["b12"],["c12"],[]],"compact_subset":["a12","b12"]}。請依 [points, compact_subset, pair_separation_rows, Hausdorff, compact, certificate] 欄序回答。

查看答案與解釋

([a12, b12, c12], [a12, b12], [[a12, b12, [[a12, c12], [b12]]], [a12, c12, [[a12, b12], [c12]]], [b12, c12, [[a12, b12], [c12]]]], 是, 是, complete_finite_topology_Hausdorff_and_finite_subset_compact)

Wave6第六層核對=([a12, b12, c12], [a12, b12], [[a12, b12, [[a12, c12], [b12]]], [a12, c12, [[a12, b12], [c12]]], [b12, c12, [[a12, b12], [c12]]]], 是, 是, complete_finite_topology_Hausdorff_and_finite_subset_compact)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。

本題逐步詳解（AI 二次覆核）

Hausdorff空間中的緊緻集為閉集 · qf-mod43-compactclosed-hypotheses-qualification · candidate

14. Hausdorff空間中的緊緻集為閉集

ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-hypotheses-qualification；資料={"points":["a13","b13","c13"],"topology":[["a13","b13","c13"],["a13","b13"],["a13","c13"],["a13"],["b13","c13"],["b13"],["c13"],[]],"compact_subset":["a13","b13"]}。請依 [points, compact_subset, pair_separation_rows, Hausdorff, compact, certificate] 欄序回答。

查看答案與解釋

([a13, b13, c13], [a13, b13], [[a13, b13, [[a13, c13], [b13]]], [a13, c13, [[a13, b13], [c13]]], [b13, c13, [[a13, b13], [c13]]]], 是, 是, complete_finite_topology_Hausdorff_and_finite_subset_compact)

Wave6第六層核對=([a13, b13, c13], [a13, b13], [[a13, b13, [[a13, c13], [b13]]], [a13, c13, [[a13, b13], [c13]]], [b13, c13, [[a13, b13], [c13]]]], 是, 是, complete_finite_topology_Hausdorff_and_finite_subset_compact)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。

本題逐步詳解（AI 二次覆核）

Hausdorff空間中的緊緻集為閉集 · qf-mod43-compactclosed-hypotheses-qualification · candidate

15. Hausdorff空間中的緊緻集為閉集

ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-hypotheses-qualification；資料={"points":["a14","b14","c14"],"topology":[["a14","b14","c14"],["a14","b14"],["a14","c14"],["a14"],["b14","c14"],["b14"],["c14"],[]],"compact_subset":["a14","b14"]}。請依 [points, compact_subset, pair_separation_rows, Hausdorff, compact, certificate] 欄序回答。

查看答案與解釋

([a14, b14, c14], [a14, b14], [[a14, b14, [[a14, c14], [b14]]], [a14, c14, [[a14, b14], [c14]]], [b14, c14, [[a14, b14], [c14]]]], 是, 是, complete_finite_topology_Hausdorff_and_finite_subset_compact)

Wave6第六層核對=([a14, b14, c14], [a14, b14], [[a14, b14, [[a14, c14], [b14]]], [a14, c14, [[a14, b14], [c14]]], [b14, c14, [[a14, b14], [c14]]]], 是, 是, complete_finite_topology_Hausdorff_and_finite_subset_compact)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。

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Hausdorff空間中的緊緻集為閉集 · qf-mod43-compactclosed-hypotheses-qualification · candidate

16. Hausdorff空間中的緊緻集為閉集

ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-hypotheses-qualification；資料={"points":["a15","b15","c15"],"topology":[["a15","b15","c15"],["a15","b15"],["a15","c15"],["a15"],["b15","c15"],["b15"],["c15"],[]],"compact_subset":["a15","b15"]}。請依 [points, compact_subset, pair_separation_rows, Hausdorff, compact, certificate] 欄序回答。

查看答案與解釋

([a15, b15, c15], [a15, b15], [[a15, b15, [[a15, c15], [b15]]], [a15, c15, [[a15, b15], [c15]]], [b15, c15, [[a15, b15], [c15]]]], 是, 是, complete_finite_topology_Hausdorff_and_finite_subset_compact)

Wave6第六層核對=([a15, b15, c15], [a15, b15], [[a15, b15, [[a15, c15], [b15]]], [a15, c15, [[a15, b15], [c15]]], [b15, c15, [[a15, b15], [c15]]]], 是, 是, complete_finite_topology_Hausdorff_and_finite_subset_compact)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。

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Hausdorff空間中的緊緻集為閉集 · qf-mod43-compactclosed-hypotheses-qualification · candidate

17. Hausdorff空間中的緊緻集為閉集

ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-hypotheses-qualification；資料={"points":["a16","b16","c16"],"topology":[["a16","b16","c16"],["a16","b16"],["a16","c16"],["a16"],["b16","c16"],["b16"],["c16"],[]],"compact_subset":["a16","b16"]}。請依 [points, compact_subset, pair_separation_rows, Hausdorff, compact, certificate] 欄序回答。

查看答案與解釋

([a16, b16, c16], [a16, b16], [[a16, b16, [[a16, c16], [b16]]], [a16, c16, [[a16, b16], [c16]]], [b16, c16, [[a16, b16], [c16]]]], 是, 是, complete_finite_topology_Hausdorff_and_finite_subset_compact)

Wave6第六層核對=([a16, b16, c16], [a16, b16], [[a16, b16, [[a16, c16], [b16]]], [a16, c16, [[a16, b16], [c16]]], [b16, c16, [[a16, b16], [c16]]]], 是, 是, complete_finite_topology_Hausdorff_and_finite_subset_compact)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。

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Hausdorff空間中的緊緻集為閉集 · qf-mod43-compactclosed-hypotheses-qualification · candidate

18. Hausdorff空間中的緊緻集為閉集

ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-hypotheses-qualification；資料={"points":["a17","b17","c17"],"topology":[["a17","b17","c17"],["a17","b17"],["a17","c17"],["a17"],["b17","c17"],["b17"],["c17"],[]],"compact_subset":["a17","b17"]}。請依 [points, compact_subset, pair_separation_rows, Hausdorff, compact, certificate] 欄序回答。

查看答案與解釋

([a17, b17, c17], [a17, b17], [[a17, b17, [[a17, c17], [b17]]], [a17, c17, [[a17, b17], [c17]]], [b17, c17, [[a17, b17], [c17]]]], 是, 是, complete_finite_topology_Hausdorff_and_finite_subset_compact)

Wave6第六層核對=([a17, b17, c17], [a17, b17], [[a17, b17, [[a17, c17], [b17]]], [a17, c17, [[a17, b17], [c17]]], [b17, c17, [[a17, b17], [c17]]]], 是, 是, complete_finite_topology_Hausdorff_and_finite_subset_compact)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。

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Hausdorff空間中的緊緻集為閉集 · qf-mod43-compactclosed-hypotheses-qualification · candidate

19. Hausdorff空間中的緊緻集為閉集

ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-hypotheses-qualification；資料={"points":["a18","b18","c18"],"topology":[["a18","b18","c18"],["a18","b18"],["a18","c18"],["a18"],["b18","c18"],["b18"],["c18"],[]],"compact_subset":["a18","b18"]}。請依 [points, compact_subset, pair_separation_rows, Hausdorff, compact, certificate] 欄序回答。

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([a18, b18, c18], [a18, b18], [[a18, b18, [[a18, c18], [b18]]], [a18, c18, [[a18, b18], [c18]]], [b18, c18, [[a18, b18], [c18]]]], 是, 是, complete_finite_topology_Hausdorff_and_finite_subset_compact)

Wave6第六層核對=([a18, b18, c18], [a18, b18], [[a18, b18, [[a18, c18], [b18]]], [a18, c18, [[a18, b18], [c18]]], [b18, c18, [[a18, b18], [c18]]]], 是, 是, complete_finite_topology_Hausdorff_and_finite_subset_compact)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。

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Hausdorff空間中的緊緻集為閉集 · qf-mod43-compactclosed-hypotheses-qualification · candidate

20. Hausdorff空間中的緊緻集為閉集

ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-hypotheses-qualification；資料={"points":["a19","b19","c19"],"topology":[["a19","b19","c19"],["a19","b19"],["a19","c19"],["a19"],["b19","c19"],["b19"],["c19"],[]],"compact_subset":["a19","b19"]}。請依 [points, compact_subset, pair_separation_rows, Hausdorff, compact, certificate] 欄序回答。

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([a19, b19, c19], [a19, b19], [[a19, b19, [[a19, c19], [b19]]], [a19, c19, [[a19, b19], [c19]]], [b19, c19, [[a19, b19], [c19]]]], 是, 是, complete_finite_topology_Hausdorff_and_finite_subset_compact)

Wave6第六層核對=([a19, b19, c19], [a19, b19], [[a19, b19, [[a19, c19], [b19]]], [a19, c19, [[a19, b19], [c19]]], [b19, c19, [[a19, b19], [c19]]]], 是, 是, complete_finite_topology_Hausdorff_and_finite_subset_compact)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。

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Hausdorff空間中的緊緻集為閉集 · qf-mod43-compactclosed-hypotheses-qualification · candidate

21. Hausdorff空間中的緊緻集為閉集

ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-hypotheses-qualification；資料={"points":["a20","b20","c20"],"topology":[["a20","b20","c20"],["a20","b20"],["a20","c20"],["a20"],["b20","c20"],["b20"],["c20"],[]],"compact_subset":["a20","b20"]}。請依 [points, compact_subset, pair_separation_rows, Hausdorff, compact, certificate] 欄序回答。

查看答案與解釋

([a20, b20, c20], [a20, b20], [[a20, b20, [[a20, c20], [b20]]], [a20, c20, [[a20, b20], [c20]]], [b20, c20, [[a20, b20], [c20]]]], 是, 是, complete_finite_topology_Hausdorff_and_finite_subset_compact)

Wave6第六層核對=([a20, b20, c20], [a20, b20], [[a20, b20, [[a20, c20], [b20]]], [a20, c20, [[a20, b20], [c20]]], [b20, c20, [[a20, b20], [c20]]]], 是, 是, complete_finite_topology_Hausdorff_and_finite_subset_compact)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。

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Hausdorff空間中的緊緻集為閉集 · qf-mod43-compactclosed-hypotheses-qualification · candidate

22. Hausdorff空間中的緊緻集為閉集

ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-hypotheses-qualification；資料={"points":["a21","b21","c21"],"topology":[["a21","b21","c21"],["a21","b21"],["a21","c21"],["a21"],["b21","c21"],["b21"],["c21"],[]],"compact_subset":["a21","b21"]}。請依 [points, compact_subset, pair_separation_rows, Hausdorff, compact, certificate] 欄序回答。

查看答案與解釋

([a21, b21, c21], [a21, b21], [[a21, b21, [[a21, c21], [b21]]], [a21, c21, [[a21, b21], [c21]]], [b21, c21, [[a21, b21], [c21]]]], 是, 是, complete_finite_topology_Hausdorff_and_finite_subset_compact)

Wave6第六層核對=([a21, b21, c21], [a21, b21], [[a21, b21, [[a21, c21], [b21]]], [a21, c21, [[a21, b21], [c21]]], [b21, c21, [[a21, b21], [c21]]]], 是, 是, complete_finite_topology_Hausdorff_and_finite_subset_compact)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。

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Hausdorff空間中的緊緻集為閉集 · qf-mod43-compactclosed-hypotheses-qualification · candidate

23. Hausdorff空間中的緊緻集為閉集

ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-hypotheses-qualification；資料={"points":["a22","b22","c22"],"topology":[["a22","b22","c22"],["a22","b22"],["a22","c22"],["a22"],["b22","c22"],["b22"],["c22"],[]],"compact_subset":["a22","b22"]}。請依 [points, compact_subset, pair_separation_rows, Hausdorff, compact, certificate] 欄序回答。

查看答案與解釋

([a22, b22, c22], [a22, b22], [[a22, b22, [[a22, c22], [b22]]], [a22, c22, [[a22, b22], [c22]]], [b22, c22, [[a22, b22], [c22]]]], 是, 是, complete_finite_topology_Hausdorff_and_finite_subset_compact)

Wave6第六層核對=([a22, b22, c22], [a22, b22], [[a22, b22, [[a22, c22], [b22]]], [a22, c22, [[a22, b22], [c22]]], [b22, c22, [[a22, b22], [c22]]]], 是, 是, complete_finite_topology_Hausdorff_and_finite_subset_compact)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。

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Hausdorff空間中的緊緻集為閉集 · qf-mod43-compactclosed-hypotheses-qualification · candidate

24. Hausdorff空間中的緊緻集為閉集

ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-hypotheses-qualification；資料={"points":["a23","b23","c23"],"topology":[["a23","b23","c23"],["a23","b23"],["a23","c23"],["a23"],["b23","c23"],["b23"],["c23"],[]],"compact_subset":["a23","b23"]}。請依 [points, compact_subset, pair_separation_rows, Hausdorff, compact, certificate] 欄序回答。

查看答案與解釋

([a23, b23, c23], [a23, b23], [[a23, b23, [[a23, c23], [b23]]], [a23, c23, [[a23, b23], [c23]]], [b23, c23, [[a23, b23], [c23]]]], 是, 是, complete_finite_topology_Hausdorff_and_finite_subset_compact)

Wave6第六層核對=([a23, b23, c23], [a23, b23], [[a23, b23, [[a23, c23], [b23]]], [a23, c23, [[a23, b23], [c23]]], [b23, c23, [[a23, b23], [c23]]]], 是, 是, complete_finite_topology_Hausdorff_and_finite_subset_compact)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。

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Hausdorff空間中的緊緻集為閉集 · qf-mod43-compactclosed-hypotheses-qualification · candidate

25. Hausdorff空間中的緊緻集為閉集

ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-hypotheses-qualification；資料={"points":["a24","b24","c24"],"topology":[["a24","b24","c24"],["a24","b24"],["a24","c24"],["a24"],["b24","c24"],["b24"],["c24"],[]],"compact_subset":["a24","b24"]}。請依 [points, compact_subset, pair_separation_rows, Hausdorff, compact, certificate] 欄序回答。

查看答案與解釋

([a24, b24, c24], [a24, b24], [[a24, b24, [[a24, c24], [b24]]], [a24, c24, [[a24, b24], [c24]]], [b24, c24, [[a24, b24], [c24]]]], 是, 是, complete_finite_topology_Hausdorff_and_finite_subset_compact)

Wave6第六層核對=([a24, b24, c24], [a24, b24], [[a24, b24, [[a24, c24], [b24]]], [a24, c24, [[a24, b24], [c24]]], [b24, c24, [[a24, b24], [c24]]]], 是, 是, complete_finite_topology_Hausdorff_and_finite_subset_compact)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。

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Hausdorff空間中的緊緻集為閉集 · qf-mod43-compactclosed-hypotheses-qualification · candidate

26. Hausdorff空間中的緊緻集為閉集

ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-finite-closedness-audit；資料={"points":["a0","b0","c0"],"topology":[["a0","b0","c0"],["a0","b0"],["a0","c0"],["a0"],["b0","c0"],["b0"],["c0"],[]],"compact_subset":["a0","b0"]}。請依 [compact_subset, complement, compact, closed, certificate] 欄序回答。

查看答案與解釋

([a0, b0], [c0], 是, 是, complement_is_open_so_compact_subset_is_closed)

Wave6第六層核對=([a0, b0], [c0], 是, 是, complement_is_open_so_compact_subset_is_closed)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。

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Hausdorff空間中的緊緻集為閉集 · qf-mod43-compactclosed-finite-closedness-audit · candidate

27. Hausdorff空間中的緊緻集為閉集

ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-finite-closedness-audit；資料={"points":["a1","b1","c1"],"topology":[["a1","b1","c1"],["a1","b1"],["a1","c1"],["a1"],["b1","c1"],["b1"],["c1"],[]],"compact_subset":["a1","b1"]}。請依 [compact_subset, complement, compact, closed, certificate] 欄序回答。

查看答案與解釋

([a1, b1], [c1], 是, 是, complement_is_open_so_compact_subset_is_closed)

Wave6第六層核對=([a1, b1], [c1], 是, 是, complement_is_open_so_compact_subset_is_closed)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。

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Hausdorff空間中的緊緻集為閉集 · qf-mod43-compactclosed-finite-closedness-audit · candidate

28. Hausdorff空間中的緊緻集為閉集

ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-finite-closedness-audit；資料={"points":["a2","b2","c2"],"topology":[["a2","b2","c2"],["a2","b2"],["a2","c2"],["a2"],["b2","c2"],["b2"],["c2"],[]],"compact_subset":["a2","b2"]}。請依 [compact_subset, complement, compact, closed, certificate] 欄序回答。

查看答案與解釋

([a2, b2], [c2], 是, 是, complement_is_open_so_compact_subset_is_closed)

Wave6第六層核對=([a2, b2], [c2], 是, 是, complement_is_open_so_compact_subset_is_closed)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。

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Hausdorff空間中的緊緻集為閉集 · qf-mod43-compactclosed-finite-closedness-audit · candidate

29. Hausdorff空間中的緊緻集為閉集

ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-finite-closedness-audit；資料={"points":["a3","b3","c3"],"topology":[["a3","b3","c3"],["a3","b3"],["a3","c3"],["a3"],["b3","c3"],["b3"],["c3"],[]],"compact_subset":["a3","b3"]}。請依 [compact_subset, complement, compact, closed, certificate] 欄序回答。

查看答案與解釋

([a3, b3], [c3], 是, 是, complement_is_open_so_compact_subset_is_closed)

Wave6第六層核對=([a3, b3], [c3], 是, 是, complement_is_open_so_compact_subset_is_closed)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。

本題逐步詳解（AI 二次覆核）

Hausdorff空間中的緊緻集為閉集 · qf-mod43-compactclosed-finite-closedness-audit · candidate

30. Hausdorff空間中的緊緻集為閉集

ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-finite-closedness-audit；資料={"points":["a4","b4","c4"],"topology":[["a4","b4","c4"],["a4","b4"],["a4","c4"],["a4"],["b4","c4"],["b4"],["c4"],[]],"compact_subset":["a4","b4"]}。請依 [compact_subset, complement, compact, closed, certificate] 欄序回答。

查看答案與解釋

([a4, b4], [c4], 是, 是, complement_is_open_so_compact_subset_is_closed)

Wave6第六層核對=([a4, b4], [c4], 是, 是, complement_is_open_so_compact_subset_is_closed)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。

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Hausdorff空間中的緊緻集為閉集 · qf-mod43-compactclosed-finite-closedness-audit · candidate

31. Hausdorff空間中的緊緻集為閉集

ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-finite-closedness-audit；資料={"points":["a5","b5","c5"],"topology":[["a5","b5","c5"],["a5","b5"],["a5","c5"],["a5"],["b5","c5"],["b5"],["c5"],[]],"compact_subset":["a5","b5"]}。請依 [compact_subset, complement, compact, closed, certificate] 欄序回答。

查看答案與解釋

([a5, b5], [c5], 是, 是, complement_is_open_so_compact_subset_is_closed)

Wave6第六層核對=([a5, b5], [c5], 是, 是, complement_is_open_so_compact_subset_is_closed)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。

本題逐步詳解（AI 二次覆核）

Hausdorff空間中的緊緻集為閉集 · qf-mod43-compactclosed-finite-closedness-audit · candidate

32. Hausdorff空間中的緊緻集為閉集

ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-finite-closedness-audit；資料={"points":["a6","b6","c6"],"topology":[["a6","b6","c6"],["a6","b6"],["a6","c6"],["a6"],["b6","c6"],["b6"],["c6"],[]],"compact_subset":["a6","b6"]}。請依 [compact_subset, complement, compact, closed, certificate] 欄序回答。

查看答案與解釋

([a6, b6], [c6], 是, 是, complement_is_open_so_compact_subset_is_closed)

Wave6第六層核對=([a6, b6], [c6], 是, 是, complement_is_open_so_compact_subset_is_closed)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。

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Hausdorff空間中的緊緻集為閉集 · qf-mod43-compactclosed-finite-closedness-audit · candidate

33. Hausdorff空間中的緊緻集為閉集

ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-finite-closedness-audit；資料={"points":["a7","b7","c7"],"topology":[["a7","b7","c7"],["a7","b7"],["a7","c7"],["a7"],["b7","c7"],["b7"],["c7"],[]],"compact_subset":["a7","b7"]}。請依 [compact_subset, complement, compact, closed, certificate] 欄序回答。

查看答案與解釋

([a7, b7], [c7], 是, 是, complement_is_open_so_compact_subset_is_closed)

Wave6第六層核對=([a7, b7], [c7], 是, 是, complement_is_open_so_compact_subset_is_closed)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。

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Hausdorff空間中的緊緻集為閉集 · qf-mod43-compactclosed-finite-closedness-audit · candidate

34. Hausdorff空間中的緊緻集為閉集

ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-finite-closedness-audit；資料={"points":["a8","b8","c8"],"topology":[["a8","b8","c8"],["a8","b8"],["a8","c8"],["a8"],["b8","c8"],["b8"],["c8"],[]],"compact_subset":["a8","b8"]}。請依 [compact_subset, complement, compact, closed, certificate] 欄序回答。

查看答案與解釋

([a8, b8], [c8], 是, 是, complement_is_open_so_compact_subset_is_closed)

Wave6第六層核對=([a8, b8], [c8], 是, 是, complement_is_open_so_compact_subset_is_closed)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。

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Hausdorff空間中的緊緻集為閉集 · qf-mod43-compactclosed-finite-closedness-audit · candidate

35. Hausdorff空間中的緊緻集為閉集

ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-finite-closedness-audit；資料={"points":["a9","b9","c9"],"topology":[["a9","b9","c9"],["a9","b9"],["a9","c9"],["a9"],["b9","c9"],["b9"],["c9"],[]],"compact_subset":["a9","b9"]}。請依 [compact_subset, complement, compact, closed, certificate] 欄序回答。

查看答案與解釋

([a9, b9], [c9], 是, 是, complement_is_open_so_compact_subset_is_closed)

Wave6第六層核對=([a9, b9], [c9], 是, 是, complement_is_open_so_compact_subset_is_closed)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。

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Hausdorff空間中的緊緻集為閉集 · qf-mod43-compactclosed-finite-closedness-audit · candidate

36. Hausdorff空間中的緊緻集為閉集

ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-finite-closedness-audit；資料={"points":["a10","b10","c10"],"topology":[["a10","b10","c10"],["a10","b10"],["a10","c10"],["a10"],["b10","c10"],["b10"],["c10"],[]],"compact_subset":["a10","b10"]}。請依 [compact_subset, complement, compact, closed, certificate] 欄序回答。

查看答案與解釋

([a10, b10], [c10], 是, 是, complement_is_open_so_compact_subset_is_closed)

Wave6第六層核對=([a10, b10], [c10], 是, 是, complement_is_open_so_compact_subset_is_closed)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。

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Hausdorff空間中的緊緻集為閉集 · qf-mod43-compactclosed-finite-closedness-audit · candidate

37. Hausdorff空間中的緊緻集為閉集

ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-finite-closedness-audit；資料={"points":["a11","b11","c11"],"topology":[["a11","b11","c11"],["a11","b11"],["a11","c11"],["a11"],["b11","c11"],["b11"],["c11"],[]],"compact_subset":["a11","b11"]}。請依 [compact_subset, complement, compact, closed, certificate] 欄序回答。

查看答案與解釋

([a11, b11], [c11], 是, 是, complement_is_open_so_compact_subset_is_closed)

Wave6第六層核對=([a11, b11], [c11], 是, 是, complement_is_open_so_compact_subset_is_closed)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。

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Hausdorff空間中的緊緻集為閉集 · qf-mod43-compactclosed-finite-closedness-audit · candidate

38. Hausdorff空間中的緊緻集為閉集

ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-finite-closedness-audit；資料={"points":["a12","b12","c12"],"topology":[["a12","b12","c12"],["a12","b12"],["a12","c12"],["a12"],["b12","c12"],["b12"],["c12"],[]],"compact_subset":["a12","b12"]}。請依 [compact_subset, complement, compact, closed, certificate] 欄序回答。

查看答案與解釋

([a12, b12], [c12], 是, 是, complement_is_open_so_compact_subset_is_closed)

Wave6第六層核對=([a12, b12], [c12], 是, 是, complement_is_open_so_compact_subset_is_closed)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。

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Hausdorff空間中的緊緻集為閉集 · qf-mod43-compactclosed-finite-closedness-audit · candidate

39. Hausdorff空間中的緊緻集為閉集

ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-finite-closedness-audit；資料={"points":["a13","b13","c13"],"topology":[["a13","b13","c13"],["a13","b13"],["a13","c13"],["a13"],["b13","c13"],["b13"],["c13"],[]],"compact_subset":["a13","b13"]}。請依 [compact_subset, complement, compact, closed, certificate] 欄序回答。

查看答案與解釋

([a13, b13], [c13], 是, 是, complement_is_open_so_compact_subset_is_closed)

Wave6第六層核對=([a13, b13], [c13], 是, 是, complement_is_open_so_compact_subset_is_closed)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。

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Hausdorff空間中的緊緻集為閉集 · qf-mod43-compactclosed-finite-closedness-audit · candidate

40. Hausdorff空間中的緊緻集為閉集

ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-finite-closedness-audit；資料={"points":["a14","b14","c14"],"topology":[["a14","b14","c14"],["a14","b14"],["a14","c14"],["a14"],["b14","c14"],["b14"],["c14"],[]],"compact_subset":["a14","b14"]}。請依 [compact_subset, complement, compact, closed, certificate] 欄序回答。

查看答案與解釋

([a14, b14], [c14], 是, 是, complement_is_open_so_compact_subset_is_closed)

Wave6第六層核對=([a14, b14], [c14], 是, 是, complement_is_open_so_compact_subset_is_closed)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。

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Hausdorff空間中的緊緻集為閉集 · qf-mod43-compactclosed-finite-closedness-audit · candidate

41. Hausdorff空間中的緊緻集為閉集

ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-finite-closedness-audit；資料={"points":["a15","b15","c15"],"topology":[["a15","b15","c15"],["a15","b15"],["a15","c15"],["a15"],["b15","c15"],["b15"],["c15"],[]],"compact_subset":["a15","b15"]}。請依 [compact_subset, complement, compact, closed, certificate] 欄序回答。

查看答案與解釋

([a15, b15], [c15], 是, 是, complement_is_open_so_compact_subset_is_closed)

Wave6第六層核對=([a15, b15], [c15], 是, 是, complement_is_open_so_compact_subset_is_closed)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。

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Hausdorff空間中的緊緻集為閉集 · qf-mod43-compactclosed-finite-closedness-audit · candidate

42. Hausdorff空間中的緊緻集為閉集

ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-finite-closedness-audit；資料={"points":["a16","b16","c16"],"topology":[["a16","b16","c16"],["a16","b16"],["a16","c16"],["a16"],["b16","c16"],["b16"],["c16"],[]],"compact_subset":["a16","b16"]}。請依 [compact_subset, complement, compact, closed, certificate] 欄序回答。

查看答案與解釋

([a16, b16], [c16], 是, 是, complement_is_open_so_compact_subset_is_closed)

Wave6第六層核對=([a16, b16], [c16], 是, 是, complement_is_open_so_compact_subset_is_closed)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。

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Hausdorff空間中的緊緻集為閉集 · qf-mod43-compactclosed-finite-closedness-audit · candidate

43. Hausdorff空間中的緊緻集為閉集

ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-finite-closedness-audit；資料={"points":["a17","b17","c17"],"topology":[["a17","b17","c17"],["a17","b17"],["a17","c17"],["a17"],["b17","c17"],["b17"],["c17"],[]],"compact_subset":["a17","b17"]}。請依 [compact_subset, complement, compact, closed, certificate] 欄序回答。

查看答案與解釋

([a17, b17], [c17], 是, 是, complement_is_open_so_compact_subset_is_closed)

Wave6第六層核對=([a17, b17], [c17], 是, 是, complement_is_open_so_compact_subset_is_closed)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。

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Hausdorff空間中的緊緻集為閉集 · qf-mod43-compactclosed-finite-closedness-audit · candidate

44. Hausdorff空間中的緊緻集為閉集

ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-finite-closedness-audit；資料={"points":["a18","b18","c18"],"topology":[["a18","b18","c18"],["a18","b18"],["a18","c18"],["a18"],["b18","c18"],["b18"],["c18"],[]],"compact_subset":["a18","b18"]}。請依 [compact_subset, complement, compact, closed, certificate] 欄序回答。

查看答案與解釋

([a18, b18], [c18], 是, 是, complement_is_open_so_compact_subset_is_closed)

Wave6第六層核對=([a18, b18], [c18], 是, 是, complement_is_open_so_compact_subset_is_closed)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。

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Hausdorff空間中的緊緻集為閉集 · qf-mod43-compactclosed-finite-closedness-audit · candidate

45. Hausdorff空間中的緊緻集為閉集

ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-finite-closedness-audit；資料={"points":["a19","b19","c19"],"topology":[["a19","b19","c19"],["a19","b19"],["a19","c19"],["a19"],["b19","c19"],["b19"],["c19"],[]],"compact_subset":["a19","b19"]}。請依 [compact_subset, complement, compact, closed, certificate] 欄序回答。

查看答案與解釋

([a19, b19], [c19], 是, 是, complement_is_open_so_compact_subset_is_closed)

Wave6第六層核對=([a19, b19], [c19], 是, 是, complement_is_open_so_compact_subset_is_closed)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。

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Hausdorff空間中的緊緻集為閉集 · qf-mod43-compactclosed-finite-closedness-audit · candidate

46. Hausdorff空間中的緊緻集為閉集

ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-finite-closedness-audit；資料={"points":["a20","b20","c20"],"topology":[["a20","b20","c20"],["a20","b20"],["a20","c20"],["a20"],["b20","c20"],["b20"],["c20"],[]],"compact_subset":["a20","b20"]}。請依 [compact_subset, complement, compact, closed, certificate] 欄序回答。

查看答案與解釋

([a20, b20], [c20], 是, 是, complement_is_open_so_compact_subset_is_closed)

Wave6第六層核對=([a20, b20], [c20], 是, 是, complement_is_open_so_compact_subset_is_closed)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。

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Hausdorff空間中的緊緻集為閉集 · qf-mod43-compactclosed-finite-closedness-audit · candidate

47. Hausdorff空間中的緊緻集為閉集

ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-finite-closedness-audit；資料={"points":["a21","b21","c21"],"topology":[["a21","b21","c21"],["a21","b21"],["a21","c21"],["a21"],["b21","c21"],["b21"],["c21"],[]],"compact_subset":["a21","b21"]}。請依 [compact_subset, complement, compact, closed, certificate] 欄序回答。

查看答案與解釋

([a21, b21], [c21], 是, 是, complement_is_open_so_compact_subset_is_closed)

Wave6第六層核對=([a21, b21], [c21], 是, 是, complement_is_open_so_compact_subset_is_closed)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。

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Hausdorff空間中的緊緻集為閉集 · qf-mod43-compactclosed-finite-closedness-audit · candidate

48. Hausdorff空間中的緊緻集為閉集

ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-finite-closedness-audit；資料={"points":["a22","b22","c22"],"topology":[["a22","b22","c22"],["a22","b22"],["a22","c22"],["a22"],["b22","c22"],["b22"],["c22"],[]],"compact_subset":["a22","b22"]}。請依 [compact_subset, complement, compact, closed, certificate] 欄序回答。

查看答案與解釋

([a22, b22], [c22], 是, 是, complement_is_open_so_compact_subset_is_closed)

Wave6第六層核對=([a22, b22], [c22], 是, 是, complement_is_open_so_compact_subset_is_closed)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。

本題逐步詳解（AI 二次覆核）

Hausdorff空間中的緊緻集為閉集 · qf-mod43-compactclosed-finite-closedness-audit · candidate

49. Hausdorff空間中的緊緻集為閉集

ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-finite-closedness-audit；資料={"points":["a23","b23","c23"],"topology":[["a23","b23","c23"],["a23","b23"],["a23","c23"],["a23"],["b23","c23"],["b23"],["c23"],[]],"compact_subset":["a23","b23"]}。請依 [compact_subset, complement, compact, closed, certificate] 欄序回答。

查看答案與解釋

([a23, b23], [c23], 是, 是, complement_is_open_so_compact_subset_is_closed)

Wave6第六層核對=([a23, b23], [c23], 是, 是, complement_is_open_so_compact_subset_is_closed)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。

本題逐步詳解（AI 二次覆核）

Hausdorff空間中的緊緻集為閉集 · qf-mod43-compactclosed-finite-closedness-audit · candidate

50. Hausdorff空間中的緊緻集為閉集

ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-finite-closedness-audit；資料={"points":["a24","b24","c24"],"topology":[["a24","b24","c24"],["a24","b24"],["a24","c24"],["a24"],["b24","c24"],["b24"],["c24"],[]],"compact_subset":["a24","b24"]}。請依 [compact_subset, complement, compact, closed, certificate] 欄序回答。

查看答案與解釋

([a24, b24], [c24], 是, 是, complement_is_open_so_compact_subset_is_closed)

Wave6第六層核對=([a24, b24], [c24], 是, 是, complement_is_open_so_compact_subset_is_closed)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。

本題逐步詳解（AI 二次覆核）

Hausdorff空間中的緊緻集為閉集 · qf-mod43-compactclosed-finite-closedness-audit · candidate

51. Hausdorff空間中的緊緻集為閉集

ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-neighborhood-subcover-certificate；資料={"points":["a0","b0","c0"],"topology":[["a0","b0","c0"],["a0","b0"],["a0","c0"],["a0"],["b0","c0"],["b0"],["c0"],[]],"compact_subset":["a0","b0"],"exterior_point":"c0"}。請依 [exterior_point, separation_rows, exterior_neighborhood, cover_union, covers_K, neighborhood_disjoint_from_K, neighborhood_open, certificate] 欄序回答。

cover_union可超出K；外側neighborhood取有限交集，並須open且與K不交。此有限見證不聲稱minimum或unique subcover。

查看答案與解釋

(c0, [[a0, [[b0, c0], [a0]]], [b0, [[a0, c0], [b0]]]], [c0], [a0, b0], 是, 是, 是, finite_subcover_and_finite_intersection_separate_x_from_K)

Wave6第六層核對=(c0, [[a0, [[b0, c0], [a0]]], [b0, [[a0, c0], [b0]]]], [c0], [a0, b0], 是, 是, 是, finite_subcover_and_finite_intersection_separate_x_from_K)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。

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Hausdorff空間中的緊緻集為閉集 · qf-mod43-compactclosed-neighborhood-subcover-certificate · candidate

52. Hausdorff空間中的緊緻集為閉集

ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-neighborhood-subcover-certificate；資料={"points":["a1","b1","c1"],"topology":[["a1","b1","c1"],["a1","b1"],["a1","c1"],["a1"],["b1","c1"],["b1"],["c1"],[]],"compact_subset":["a1","b1"],"exterior_point":"c1"}。請依 [exterior_point, separation_rows, exterior_neighborhood, cover_union, covers_K, neighborhood_disjoint_from_K, neighborhood_open, certificate] 欄序回答。

cover_union可超出K；外側neighborhood取有限交集，並須open且與K不交。此有限見證不聲稱minimum或unique subcover。

查看答案與解釋

(c1, [[a1, [[b1, c1], [a1]]], [b1, [[a1, c1], [b1]]]], [c1], [a1, b1], 是, 是, 是, finite_subcover_and_finite_intersection_separate_x_from_K)

Wave6第六層核對=(c1, [[a1, [[b1, c1], [a1]]], [b1, [[a1, c1], [b1]]]], [c1], [a1, b1], 是, 是, 是, finite_subcover_and_finite_intersection_separate_x_from_K)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。

本題逐步詳解（AI 二次覆核）

Hausdorff空間中的緊緻集為閉集 · qf-mod43-compactclosed-neighborhood-subcover-certificate · candidate

53. Hausdorff空間中的緊緻集為閉集

ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-neighborhood-subcover-certificate；資料={"points":["a2","b2","c2"],"topology":[["a2","b2","c2"],["a2","b2"],["a2","c2"],["a2"],["b2","c2"],["b2"],["c2"],[]],"compact_subset":["a2","b2"],"exterior_point":"c2"}。請依 [exterior_point, separation_rows, exterior_neighborhood, cover_union, covers_K, neighborhood_disjoint_from_K, neighborhood_open, certificate] 欄序回答。

cover_union可超出K；外側neighborhood取有限交集，並須open且與K不交。此有限見證不聲稱minimum或unique subcover。

查看答案與解釋

(c2, [[a2, [[b2, c2], [a2]]], [b2, [[a2, c2], [b2]]]], [c2], [a2, b2], 是, 是, 是, finite_subcover_and_finite_intersection_separate_x_from_K)

Wave6第六層核對=(c2, [[a2, [[b2, c2], [a2]]], [b2, [[a2, c2], [b2]]]], [c2], [a2, b2], 是, 是, 是, finite_subcover_and_finite_intersection_separate_x_from_K)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。

本題逐步詳解（AI 二次覆核）

Hausdorff空間中的緊緻集為閉集 · qf-mod43-compactclosed-neighborhood-subcover-certificate · candidate

54. Hausdorff空間中的緊緻集為閉集

ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-neighborhood-subcover-certificate；資料={"points":["a3","b3","c3"],"topology":[["a3","b3","c3"],["a3","b3"],["a3","c3"],["a3"],["b3","c3"],["b3"],["c3"],[]],"compact_subset":["a3","b3"],"exterior_point":"c3"}。請依 [exterior_point, separation_rows, exterior_neighborhood, cover_union, covers_K, neighborhood_disjoint_from_K, neighborhood_open, certificate] 欄序回答。

cover_union可超出K；外側neighborhood取有限交集，並須open且與K不交。此有限見證不聲稱minimum或unique subcover。

查看答案與解釋

(c3, [[a3, [[b3, c3], [a3]]], [b3, [[a3, c3], [b3]]]], [c3], [a3, b3], 是, 是, 是, finite_subcover_and_finite_intersection_separate_x_from_K)

Wave6第六層核對=(c3, [[a3, [[b3, c3], [a3]]], [b3, [[a3, c3], [b3]]]], [c3], [a3, b3], 是, 是, 是, finite_subcover_and_finite_intersection_separate_x_from_K)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。

本題逐步詳解（AI 二次覆核）

Hausdorff空間中的緊緻集為閉集 · qf-mod43-compactclosed-neighborhood-subcover-certificate · candidate

55. Hausdorff空間中的緊緻集為閉集

ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-neighborhood-subcover-certificate；資料={"points":["a4","b4","c4"],"topology":[["a4","b4","c4"],["a4","b4"],["a4","c4"],["a4"],["b4","c4"],["b4"],["c4"],[]],"compact_subset":["a4","b4"],"exterior_point":"c4"}。請依 [exterior_point, separation_rows, exterior_neighborhood, cover_union, covers_K, neighborhood_disjoint_from_K, neighborhood_open, certificate] 欄序回答。

cover_union可超出K；外側neighborhood取有限交集，並須open且與K不交。此有限見證不聲稱minimum或unique subcover。

查看答案與解釋

(c4, [[a4, [[b4, c4], [a4]]], [b4, [[a4, c4], [b4]]]], [c4], [a4, b4], 是, 是, 是, finite_subcover_and_finite_intersection_separate_x_from_K)

Wave6第六層核對=(c4, [[a4, [[b4, c4], [a4]]], [b4, [[a4, c4], [b4]]]], [c4], [a4, b4], 是, 是, 是, finite_subcover_and_finite_intersection_separate_x_from_K)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。

本題逐步詳解（AI 二次覆核）

Hausdorff空間中的緊緻集為閉集 · qf-mod43-compactclosed-neighborhood-subcover-certificate · candidate

56. Hausdorff空間中的緊緻集為閉集

ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-neighborhood-subcover-certificate；資料={"points":["a5","b5","c5"],"topology":[["a5","b5","c5"],["a5","b5"],["a5","c5"],["a5"],["b5","c5"],["b5"],["c5"],[]],"compact_subset":["a5","b5"],"exterior_point":"c5"}。請依 [exterior_point, separation_rows, exterior_neighborhood, cover_union, covers_K, neighborhood_disjoint_from_K, neighborhood_open, certificate] 欄序回答。

cover_union可超出K；外側neighborhood取有限交集，並須open且與K不交。此有限見證不聲稱minimum或unique subcover。

查看答案與解釋

(c5, [[a5, [[b5, c5], [a5]]], [b5, [[a5, c5], [b5]]]], [c5], [a5, b5], 是, 是, 是, finite_subcover_and_finite_intersection_separate_x_from_K)

Wave6第六層核對=(c5, [[a5, [[b5, c5], [a5]]], [b5, [[a5, c5], [b5]]]], [c5], [a5, b5], 是, 是, 是, finite_subcover_and_finite_intersection_separate_x_from_K)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。

本題逐步詳解（AI 二次覆核）

Hausdorff空間中的緊緻集為閉集 · qf-mod43-compactclosed-neighborhood-subcover-certificate · candidate

57. Hausdorff空間中的緊緻集為閉集

ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-neighborhood-subcover-certificate；資料={"points":["a6","b6","c6"],"topology":[["a6","b6","c6"],["a6","b6"],["a6","c6"],["a6"],["b6","c6"],["b6"],["c6"],[]],"compact_subset":["a6","b6"],"exterior_point":"c6"}。請依 [exterior_point, separation_rows, exterior_neighborhood, cover_union, covers_K, neighborhood_disjoint_from_K, neighborhood_open, certificate] 欄序回答。

cover_union可超出K；外側neighborhood取有限交集，並須open且與K不交。此有限見證不聲稱minimum或unique subcover。

查看答案與解釋

(c6, [[a6, [[b6, c6], [a6]]], [b6, [[a6, c6], [b6]]]], [c6], [a6, b6], 是, 是, 是, finite_subcover_and_finite_intersection_separate_x_from_K)

Wave6第六層核對=(c6, [[a6, [[b6, c6], [a6]]], [b6, [[a6, c6], [b6]]]], [c6], [a6, b6], 是, 是, 是, finite_subcover_and_finite_intersection_separate_x_from_K)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。

本題逐步詳解（AI 二次覆核）

Hausdorff空間中的緊緻集為閉集 · qf-mod43-compactclosed-neighborhood-subcover-certificate · candidate

58. Hausdorff空間中的緊緻集為閉集

ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-neighborhood-subcover-certificate；資料={"points":["a7","b7","c7"],"topology":[["a7","b7","c7"],["a7","b7"],["a7","c7"],["a7"],["b7","c7"],["b7"],["c7"],[]],"compact_subset":["a7","b7"],"exterior_point":"c7"}。請依 [exterior_point, separation_rows, exterior_neighborhood, cover_union, covers_K, neighborhood_disjoint_from_K, neighborhood_open, certificate] 欄序回答。

cover_union可超出K；外側neighborhood取有限交集，並須open且與K不交。此有限見證不聲稱minimum或unique subcover。

查看答案與解釋

(c7, [[a7, [[b7, c7], [a7]]], [b7, [[a7, c7], [b7]]]], [c7], [a7, b7], 是, 是, 是, finite_subcover_and_finite_intersection_separate_x_from_K)

Wave6第六層核對=(c7, [[a7, [[b7, c7], [a7]]], [b7, [[a7, c7], [b7]]]], [c7], [a7, b7], 是, 是, 是, finite_subcover_and_finite_intersection_separate_x_from_K)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。

本題逐步詳解（AI 二次覆核）

Hausdorff空間中的緊緻集為閉集 · qf-mod43-compactclosed-neighborhood-subcover-certificate · candidate

59. Hausdorff空間中的緊緻集為閉集

ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-neighborhood-subcover-certificate；資料={"points":["a8","b8","c8"],"topology":[["a8","b8","c8"],["a8","b8"],["a8","c8"],["a8"],["b8","c8"],["b8"],["c8"],[]],"compact_subset":["a8","b8"],"exterior_point":"c8"}。請依 [exterior_point, separation_rows, exterior_neighborhood, cover_union, covers_K, neighborhood_disjoint_from_K, neighborhood_open, certificate] 欄序回答。

cover_union可超出K；外側neighborhood取有限交集，並須open且與K不交。此有限見證不聲稱minimum或unique subcover。

查看答案與解釋

(c8, [[a8, [[b8, c8], [a8]]], [b8, [[a8, c8], [b8]]]], [c8], [a8, b8], 是, 是, 是, finite_subcover_and_finite_intersection_separate_x_from_K)

Wave6第六層核對=(c8, [[a8, [[b8, c8], [a8]]], [b8, [[a8, c8], [b8]]]], [c8], [a8, b8], 是, 是, 是, finite_subcover_and_finite_intersection_separate_x_from_K)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。

本題逐步詳解（AI 二次覆核）

Hausdorff空間中的緊緻集為閉集 · qf-mod43-compactclosed-neighborhood-subcover-certificate · candidate

60. Hausdorff空間中的緊緻集為閉集

ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-neighborhood-subcover-certificate；資料={"points":["a9","b9","c9"],"topology":[["a9","b9","c9"],["a9","b9"],["a9","c9"],["a9"],["b9","c9"],["b9"],["c9"],[]],"compact_subset":["a9","b9"],"exterior_point":"c9"}。請依 [exterior_point, separation_rows, exterior_neighborhood, cover_union, covers_K, neighborhood_disjoint_from_K, neighborhood_open, certificate] 欄序回答。

cover_union可超出K；外側neighborhood取有限交集，並須open且與K不交。此有限見證不聲稱minimum或unique subcover。

查看答案與解釋

(c9, [[a9, [[b9, c9], [a9]]], [b9, [[a9, c9], [b9]]]], [c9], [a9, b9], 是, 是, 是, finite_subcover_and_finite_intersection_separate_x_from_K)

Wave6第六層核對=(c9, [[a9, [[b9, c9], [a9]]], [b9, [[a9, c9], [b9]]]], [c9], [a9, b9], 是, 是, 是, finite_subcover_and_finite_intersection_separate_x_from_K)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。

本題逐步詳解（AI 二次覆核）

Hausdorff空間中的緊緻集為閉集 · qf-mod43-compactclosed-neighborhood-subcover-certificate · candidate

61. Hausdorff空間中的緊緻集為閉集

ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-neighborhood-subcover-certificate；資料={"points":["a10","b10","c10"],"topology":[["a10","b10","c10"],["a10","b10"],["a10","c10"],["a10"],["b10","c10"],["b10"],["c10"],[]],"compact_subset":["a10","b10"],"exterior_point":"c10"}。請依 [exterior_point, separation_rows, exterior_neighborhood, cover_union, covers_K, neighborhood_disjoint_from_K, neighborhood_open, certificate] 欄序回答。

cover_union可超出K；外側neighborhood取有限交集，並須open且與K不交。此有限見證不聲稱minimum或unique subcover。

查看答案與解釋

(c10, [[a10, [[b10, c10], [a10]]], [b10, [[a10, c10], [b10]]]], [c10], [a10, b10], 是, 是, 是, finite_subcover_and_finite_intersection_separate_x_from_K)

Wave6第六層核對=(c10, [[a10, [[b10, c10], [a10]]], [b10, [[a10, c10], [b10]]]], [c10], [a10, b10], 是, 是, 是, finite_subcover_and_finite_intersection_separate_x_from_K)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。

本題逐步詳解（AI 二次覆核）

Hausdorff空間中的緊緻集為閉集 · qf-mod43-compactclosed-neighborhood-subcover-certificate · candidate

62. Hausdorff空間中的緊緻集為閉集

ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-neighborhood-subcover-certificate；資料={"points":["a11","b11","c11"],"topology":[["a11","b11","c11"],["a11","b11"],["a11","c11"],["a11"],["b11","c11"],["b11"],["c11"],[]],"compact_subset":["a11","b11"],"exterior_point":"c11"}。請依 [exterior_point, separation_rows, exterior_neighborhood, cover_union, covers_K, neighborhood_disjoint_from_K, neighborhood_open, certificate] 欄序回答。

cover_union可超出K；外側neighborhood取有限交集，並須open且與K不交。此有限見證不聲稱minimum或unique subcover。

查看答案與解釋

(c11, [[a11, [[b11, c11], [a11]]], [b11, [[a11, c11], [b11]]]], [c11], [a11, b11], 是, 是, 是, finite_subcover_and_finite_intersection_separate_x_from_K)

Wave6第六層核對=(c11, [[a11, [[b11, c11], [a11]]], [b11, [[a11, c11], [b11]]]], [c11], [a11, b11], 是, 是, 是, finite_subcover_and_finite_intersection_separate_x_from_K)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。

本題逐步詳解（AI 二次覆核）

Hausdorff空間中的緊緻集為閉集 · qf-mod43-compactclosed-neighborhood-subcover-certificate · candidate

63. Hausdorff空間中的緊緻集為閉集

ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-neighborhood-subcover-certificate；資料={"points":["a12","b12","c12"],"topology":[["a12","b12","c12"],["a12","b12"],["a12","c12"],["a12"],["b12","c12"],["b12"],["c12"],[]],"compact_subset":["a12","b12"],"exterior_point":"c12"}。請依 [exterior_point, separation_rows, exterior_neighborhood, cover_union, covers_K, neighborhood_disjoint_from_K, neighborhood_open, certificate] 欄序回答。

cover_union可超出K；外側neighborhood取有限交集，並須open且與K不交。此有限見證不聲稱minimum或unique subcover。

查看答案與解釋

(c12, [[a12, [[b12, c12], [a12]]], [b12, [[a12, c12], [b12]]]], [c12], [a12, b12], 是, 是, 是, finite_subcover_and_finite_intersection_separate_x_from_K)

Wave6第六層核對=(c12, [[a12, [[b12, c12], [a12]]], [b12, [[a12, c12], [b12]]]], [c12], [a12, b12], 是, 是, 是, finite_subcover_and_finite_intersection_separate_x_from_K)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。

本題逐步詳解（AI 二次覆核）

Hausdorff空間中的緊緻集為閉集 · qf-mod43-compactclosed-neighborhood-subcover-certificate · candidate

64. Hausdorff空間中的緊緻集為閉集

ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-neighborhood-subcover-certificate；資料={"points":["a13","b13","c13"],"topology":[["a13","b13","c13"],["a13","b13"],["a13","c13"],["a13"],["b13","c13"],["b13"],["c13"],[]],"compact_subset":["a13","b13"],"exterior_point":"c13"}。請依 [exterior_point, separation_rows, exterior_neighborhood, cover_union, covers_K, neighborhood_disjoint_from_K, neighborhood_open, certificate] 欄序回答。

cover_union可超出K；外側neighborhood取有限交集，並須open且與K不交。此有限見證不聲稱minimum或unique subcover。

查看答案與解釋

(c13, [[a13, [[b13, c13], [a13]]], [b13, [[a13, c13], [b13]]]], [c13], [a13, b13], 是, 是, 是, finite_subcover_and_finite_intersection_separate_x_from_K)

Wave6第六層核對=(c13, [[a13, [[b13, c13], [a13]]], [b13, [[a13, c13], [b13]]]], [c13], [a13, b13], 是, 是, 是, finite_subcover_and_finite_intersection_separate_x_from_K)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。

本題逐步詳解（AI 二次覆核）

Hausdorff空間中的緊緻集為閉集 · qf-mod43-compactclosed-neighborhood-subcover-certificate · candidate

65. Hausdorff空間中的緊緻集為閉集

ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-neighborhood-subcover-certificate；資料={"points":["a14","b14","c14"],"topology":[["a14","b14","c14"],["a14","b14"],["a14","c14"],["a14"],["b14","c14"],["b14"],["c14"],[]],"compact_subset":["a14","b14"],"exterior_point":"c14"}。請依 [exterior_point, separation_rows, exterior_neighborhood, cover_union, covers_K, neighborhood_disjoint_from_K, neighborhood_open, certificate] 欄序回答。

cover_union可超出K；外側neighborhood取有限交集，並須open且與K不交。此有限見證不聲稱minimum或unique subcover。

查看答案與解釋

(c14, [[a14, [[b14, c14], [a14]]], [b14, [[a14, c14], [b14]]]], [c14], [a14, b14], 是, 是, 是, finite_subcover_and_finite_intersection_separate_x_from_K)

Wave6第六層核對=(c14, [[a14, [[b14, c14], [a14]]], [b14, [[a14, c14], [b14]]]], [c14], [a14, b14], 是, 是, 是, finite_subcover_and_finite_intersection_separate_x_from_K)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。

本題逐步詳解（AI 二次覆核）

Hausdorff空間中的緊緻集為閉集 · qf-mod43-compactclosed-neighborhood-subcover-certificate · candidate

66. Hausdorff空間中的緊緻集為閉集

ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-neighborhood-subcover-certificate；資料={"points":["a15","b15","c15"],"topology":[["a15","b15","c15"],["a15","b15"],["a15","c15"],["a15"],["b15","c15"],["b15"],["c15"],[]],"compact_subset":["a15","b15"],"exterior_point":"c15"}。請依 [exterior_point, separation_rows, exterior_neighborhood, cover_union, covers_K, neighborhood_disjoint_from_K, neighborhood_open, certificate] 欄序回答。

cover_union可超出K；外側neighborhood取有限交集，並須open且與K不交。此有限見證不聲稱minimum或unique subcover。

查看答案與解釋

(c15, [[a15, [[b15, c15], [a15]]], [b15, [[a15, c15], [b15]]]], [c15], [a15, b15], 是, 是, 是, finite_subcover_and_finite_intersection_separate_x_from_K)

Wave6第六層核對=(c15, [[a15, [[b15, c15], [a15]]], [b15, [[a15, c15], [b15]]]], [c15], [a15, b15], 是, 是, 是, finite_subcover_and_finite_intersection_separate_x_from_K)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。

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Hausdorff空間中的緊緻集為閉集 · qf-mod43-compactclosed-neighborhood-subcover-certificate · candidate

67. Hausdorff空間中的緊緻集為閉集

ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-neighborhood-subcover-certificate；資料={"points":["a16","b16","c16"],"topology":[["a16","b16","c16"],["a16","b16"],["a16","c16"],["a16"],["b16","c16"],["b16"],["c16"],[]],"compact_subset":["a16","b16"],"exterior_point":"c16"}。請依 [exterior_point, separation_rows, exterior_neighborhood, cover_union, covers_K, neighborhood_disjoint_from_K, neighborhood_open, certificate] 欄序回答。

cover_union可超出K；外側neighborhood取有限交集，並須open且與K不交。此有限見證不聲稱minimum或unique subcover。

查看答案與解釋

(c16, [[a16, [[b16, c16], [a16]]], [b16, [[a16, c16], [b16]]]], [c16], [a16, b16], 是, 是, 是, finite_subcover_and_finite_intersection_separate_x_from_K)

Wave6第六層核對=(c16, [[a16, [[b16, c16], [a16]]], [b16, [[a16, c16], [b16]]]], [c16], [a16, b16], 是, 是, 是, finite_subcover_and_finite_intersection_separate_x_from_K)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。

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Hausdorff空間中的緊緻集為閉集 · qf-mod43-compactclosed-neighborhood-subcover-certificate · candidate

68. Hausdorff空間中的緊緻集為閉集

ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-neighborhood-subcover-certificate；資料={"points":["a17","b17","c17"],"topology":[["a17","b17","c17"],["a17","b17"],["a17","c17"],["a17"],["b17","c17"],["b17"],["c17"],[]],"compact_subset":["a17","b17"],"exterior_point":"c17"}。請依 [exterior_point, separation_rows, exterior_neighborhood, cover_union, covers_K, neighborhood_disjoint_from_K, neighborhood_open, certificate] 欄序回答。

cover_union可超出K；外側neighborhood取有限交集，並須open且與K不交。此有限見證不聲稱minimum或unique subcover。

查看答案與解釋

(c17, [[a17, [[b17, c17], [a17]]], [b17, [[a17, c17], [b17]]]], [c17], [a17, b17], 是, 是, 是, finite_subcover_and_finite_intersection_separate_x_from_K)

Wave6第六層核對=(c17, [[a17, [[b17, c17], [a17]]], [b17, [[a17, c17], [b17]]]], [c17], [a17, b17], 是, 是, 是, finite_subcover_and_finite_intersection_separate_x_from_K)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。

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Hausdorff空間中的緊緻集為閉集 · qf-mod43-compactclosed-neighborhood-subcover-certificate · candidate

69. Hausdorff空間中的緊緻集為閉集

ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-neighborhood-subcover-certificate；資料={"points":["a18","b18","c18"],"topology":[["a18","b18","c18"],["a18","b18"],["a18","c18"],["a18"],["b18","c18"],["b18"],["c18"],[]],"compact_subset":["a18","b18"],"exterior_point":"c18"}。請依 [exterior_point, separation_rows, exterior_neighborhood, cover_union, covers_K, neighborhood_disjoint_from_K, neighborhood_open, certificate] 欄序回答。

cover_union可超出K；外側neighborhood取有限交集，並須open且與K不交。此有限見證不聲稱minimum或unique subcover。

查看答案與解釋

(c18, [[a18, [[b18, c18], [a18]]], [b18, [[a18, c18], [b18]]]], [c18], [a18, b18], 是, 是, 是, finite_subcover_and_finite_intersection_separate_x_from_K)

Wave6第六層核對=(c18, [[a18, [[b18, c18], [a18]]], [b18, [[a18, c18], [b18]]]], [c18], [a18, b18], 是, 是, 是, finite_subcover_and_finite_intersection_separate_x_from_K)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。

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Hausdorff空間中的緊緻集為閉集 · qf-mod43-compactclosed-neighborhood-subcover-certificate · candidate

70. Hausdorff空間中的緊緻集為閉集

ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-neighborhood-subcover-certificate；資料={"points":["a19","b19","c19"],"topology":[["a19","b19","c19"],["a19","b19"],["a19","c19"],["a19"],["b19","c19"],["b19"],["c19"],[]],"compact_subset":["a19","b19"],"exterior_point":"c19"}。請依 [exterior_point, separation_rows, exterior_neighborhood, cover_union, covers_K, neighborhood_disjoint_from_K, neighborhood_open, certificate] 欄序回答。

cover_union可超出K；外側neighborhood取有限交集，並須open且與K不交。此有限見證不聲稱minimum或unique subcover。

查看答案與解釋

(c19, [[a19, [[b19, c19], [a19]]], [b19, [[a19, c19], [b19]]]], [c19], [a19, b19], 是, 是, 是, finite_subcover_and_finite_intersection_separate_x_from_K)

Wave6第六層核對=(c19, [[a19, [[b19, c19], [a19]]], [b19, [[a19, c19], [b19]]]], [c19], [a19, b19], 是, 是, 是, finite_subcover_and_finite_intersection_separate_x_from_K)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。

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Hausdorff空間中的緊緻集為閉集 · qf-mod43-compactclosed-neighborhood-subcover-certificate · candidate

71. Hausdorff空間中的緊緻集為閉集

ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-neighborhood-subcover-certificate；資料={"points":["a20","b20","c20"],"topology":[["a20","b20","c20"],["a20","b20"],["a20","c20"],["a20"],["b20","c20"],["b20"],["c20"],[]],"compact_subset":["a20","b20"],"exterior_point":"c20"}。請依 [exterior_point, separation_rows, exterior_neighborhood, cover_union, covers_K, neighborhood_disjoint_from_K, neighborhood_open, certificate] 欄序回答。

cover_union可超出K；外側neighborhood取有限交集，並須open且與K不交。此有限見證不聲稱minimum或unique subcover。

查看答案與解釋

(c20, [[a20, [[b20, c20], [a20]]], [b20, [[a20, c20], [b20]]]], [c20], [a20, b20], 是, 是, 是, finite_subcover_and_finite_intersection_separate_x_from_K)

Wave6第六層核對=(c20, [[a20, [[b20, c20], [a20]]], [b20, [[a20, c20], [b20]]]], [c20], [a20, b20], 是, 是, 是, finite_subcover_and_finite_intersection_separate_x_from_K)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。

本題逐步詳解（AI 二次覆核）

Hausdorff空間中的緊緻集為閉集 · qf-mod43-compactclosed-neighborhood-subcover-certificate · candidate

72. Hausdorff空間中的緊緻集為閉集

ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-neighborhood-subcover-certificate；資料={"points":["a21","b21","c21"],"topology":[["a21","b21","c21"],["a21","b21"],["a21","c21"],["a21"],["b21","c21"],["b21"],["c21"],[]],"compact_subset":["a21","b21"],"exterior_point":"c21"}。請依 [exterior_point, separation_rows, exterior_neighborhood, cover_union, covers_K, neighborhood_disjoint_from_K, neighborhood_open, certificate] 欄序回答。

cover_union可超出K；外側neighborhood取有限交集，並須open且與K不交。此有限見證不聲稱minimum或unique subcover。

查看答案與解釋

(c21, [[a21, [[b21, c21], [a21]]], [b21, [[a21, c21], [b21]]]], [c21], [a21, b21], 是, 是, 是, finite_subcover_and_finite_intersection_separate_x_from_K)

Wave6第六層核對=(c21, [[a21, [[b21, c21], [a21]]], [b21, [[a21, c21], [b21]]]], [c21], [a21, b21], 是, 是, 是, finite_subcover_and_finite_intersection_separate_x_from_K)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。

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Hausdorff空間中的緊緻集為閉集 · qf-mod43-compactclosed-neighborhood-subcover-certificate · candidate

73. Hausdorff空間中的緊緻集為閉集

ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-neighborhood-subcover-certificate；資料={"points":["a22","b22","c22"],"topology":[["a22","b22","c22"],["a22","b22"],["a22","c22"],["a22"],["b22","c22"],["b22"],["c22"],[]],"compact_subset":["a22","b22"],"exterior_point":"c22"}。請依 [exterior_point, separation_rows, exterior_neighborhood, cover_union, covers_K, neighborhood_disjoint_from_K, neighborhood_open, certificate] 欄序回答。

cover_union可超出K；外側neighborhood取有限交集，並須open且與K不交。此有限見證不聲稱minimum或unique subcover。

查看答案與解釋

(c22, [[a22, [[b22, c22], [a22]]], [b22, [[a22, c22], [b22]]]], [c22], [a22, b22], 是, 是, 是, finite_subcover_and_finite_intersection_separate_x_from_K)

Wave6第六層核對=(c22, [[a22, [[b22, c22], [a22]]], [b22, [[a22, c22], [b22]]]], [c22], [a22, b22], 是, 是, 是, finite_subcover_and_finite_intersection_separate_x_from_K)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。

本題逐步詳解（AI 二次覆核）

Hausdorff空間中的緊緻集為閉集 · qf-mod43-compactclosed-neighborhood-subcover-certificate · candidate

74. Hausdorff空間中的緊緻集為閉集

ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-neighborhood-subcover-certificate；資料={"points":["a23","b23","c23"],"topology":[["a23","b23","c23"],["a23","b23"],["a23","c23"],["a23"],["b23","c23"],["b23"],["c23"],[]],"compact_subset":["a23","b23"],"exterior_point":"c23"}。請依 [exterior_point, separation_rows, exterior_neighborhood, cover_union, covers_K, neighborhood_disjoint_from_K, neighborhood_open, certificate] 欄序回答。

cover_union可超出K；外側neighborhood取有限交集，並須open且與K不交。此有限見證不聲稱minimum或unique subcover。

查看答案與解釋

(c23, [[a23, [[b23, c23], [a23]]], [b23, [[a23, c23], [b23]]]], [c23], [a23, b23], 是, 是, 是, finite_subcover_and_finite_intersection_separate_x_from_K)

Wave6第六層核對=(c23, [[a23, [[b23, c23], [a23]]], [b23, [[a23, c23], [b23]]]], [c23], [a23, b23], 是, 是, 是, finite_subcover_and_finite_intersection_separate_x_from_K)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。

本題逐步詳解（AI 二次覆核）

Hausdorff空間中的緊緻集為閉集 · qf-mod43-compactclosed-neighborhood-subcover-certificate · candidate

75. Hausdorff空間中的緊緻集為閉集

ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-neighborhood-subcover-certificate；資料={"points":["a24","b24","c24"],"topology":[["a24","b24","c24"],["a24","b24"],["a24","c24"],["a24"],["b24","c24"],["b24"],["c24"],[]],"compact_subset":["a24","b24"],"exterior_point":"c24"}。請依 [exterior_point, separation_rows, exterior_neighborhood, cover_union, covers_K, neighborhood_disjoint_from_K, neighborhood_open, certificate] 欄序回答。

cover_union可超出K；外側neighborhood取有限交集，並須open且與K不交。此有限見證不聲稱minimum或unique subcover。

查看答案與解釋

(c24, [[a24, [[b24, c24], [a24]]], [b24, [[a24, c24], [b24]]]], [c24], [a24, b24], 是, 是, 是, finite_subcover_and_finite_intersection_separate_x_from_K)

Wave6第六層核對=(c24, [[a24, [[b24, c24], [a24]]], [b24, [[a24, c24], [b24]]]], [c24], [a24, b24], 是, 是, 是, finite_subcover_and_finite_intersection_separate_x_from_K)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。

本題逐步詳解（AI 二次覆核）

Hausdorff空間中的緊緻集為閉集 · qf-mod43-compactclosed-neighborhood-subcover-certificate · candidate

76. Hausdorff空間中的緊緻集為閉集

ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-nonhausdorff-boundary；資料={"points":["o0","u0"],"topology":[["o0","u0"],["u0"],[]],"compact_subset":["u0"]}。請依 [points, compact_subset, complement, Hausdorff, compact, closed, boundary] 欄序回答。

查看答案與解釋

([o0, u0], [u0], [o0], 否, 是, 否, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)

Wave6第六層核對=([o0, u0], [u0], [o0], 否, 是, 否, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。

本題逐步詳解（AI 二次覆核）

Hausdorff空間中的緊緻集為閉集 · qf-mod43-compactclosed-nonhausdorff-boundary · candidate

77. Hausdorff空間中的緊緻集為閉集

ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-nonhausdorff-boundary；資料={"points":["o1","u1"],"topology":[["o1","u1"],["u1"],[]],"compact_subset":["u1"]}。請依 [points, compact_subset, complement, Hausdorff, compact, closed, boundary] 欄序回答。

查看答案與解釋

([o1, u1], [u1], [o1], 否, 是, 否, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)

Wave6第六層核對=([o1, u1], [u1], [o1], 否, 是, 否, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。

本題逐步詳解（AI 二次覆核）

Hausdorff空間中的緊緻集為閉集 · qf-mod43-compactclosed-nonhausdorff-boundary · candidate

78. Hausdorff空間中的緊緻集為閉集

ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-nonhausdorff-boundary；資料={"points":["o2","u2"],"topology":[["o2","u2"],["u2"],[]],"compact_subset":["u2"]}。請依 [points, compact_subset, complement, Hausdorff, compact, closed, boundary] 欄序回答。

查看答案與解釋

([o2, u2], [u2], [o2], 否, 是, 否, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)

Wave6第六層核對=([o2, u2], [u2], [o2], 否, 是, 否, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。

本題逐步詳解（AI 二次覆核）

Hausdorff空間中的緊緻集為閉集 · qf-mod43-compactclosed-nonhausdorff-boundary · candidate

79. Hausdorff空間中的緊緻集為閉集

ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-nonhausdorff-boundary；資料={"points":["o3","u3"],"topology":[["o3","u3"],["u3"],[]],"compact_subset":["u3"]}。請依 [points, compact_subset, complement, Hausdorff, compact, closed, boundary] 欄序回答。

查看答案與解釋

([o3, u3], [u3], [o3], 否, 是, 否, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)

Wave6第六層核對=([o3, u3], [u3], [o3], 否, 是, 否, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。

本題逐步詳解（AI 二次覆核）

Hausdorff空間中的緊緻集為閉集 · qf-mod43-compactclosed-nonhausdorff-boundary · candidate

80. Hausdorff空間中的緊緻集為閉集

ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-nonhausdorff-boundary；資料={"points":["o4","u4"],"topology":[["o4","u4"],["u4"],[]],"compact_subset":["u4"]}。請依 [points, compact_subset, complement, Hausdorff, compact, closed, boundary] 欄序回答。

查看答案與解釋

([o4, u4], [u4], [o4], 否, 是, 否, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)

Wave6第六層核對=([o4, u4], [u4], [o4], 否, 是, 否, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。

本題逐步詳解（AI 二次覆核）

Hausdorff空間中的緊緻集為閉集 · qf-mod43-compactclosed-nonhausdorff-boundary · candidate

81. Hausdorff空間中的緊緻集為閉集

ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-nonhausdorff-boundary；資料={"points":["o5","u5"],"topology":[["o5","u5"],["u5"],[]],"compact_subset":["u5"]}。請依 [points, compact_subset, complement, Hausdorff, compact, closed, boundary] 欄序回答。

查看答案與解釋

([o5, u5], [u5], [o5], 否, 是, 否, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)

Wave6第六層核對=([o5, u5], [u5], [o5], 否, 是, 否, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。

本題逐步詳解（AI 二次覆核）

Hausdorff空間中的緊緻集為閉集 · qf-mod43-compactclosed-nonhausdorff-boundary · candidate

82. Hausdorff空間中的緊緻集為閉集

ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-nonhausdorff-boundary；資料={"points":["o6","u6"],"topology":[["o6","u6"],["u6"],[]],"compact_subset":["u6"]}。請依 [points, compact_subset, complement, Hausdorff, compact, closed, boundary] 欄序回答。

查看答案與解釋

([o6, u6], [u6], [o6], 否, 是, 否, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)

Wave6第六層核對=([o6, u6], [u6], [o6], 否, 是, 否, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。

本題逐步詳解（AI 二次覆核）

Hausdorff空間中的緊緻集為閉集 · qf-mod43-compactclosed-nonhausdorff-boundary · candidate

83. Hausdorff空間中的緊緻集為閉集

ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-nonhausdorff-boundary；資料={"points":["o7","u7"],"topology":[["o7","u7"],["u7"],[]],"compact_subset":["u7"]}。請依 [points, compact_subset, complement, Hausdorff, compact, closed, boundary] 欄序回答。

查看答案與解釋

([o7, u7], [u7], [o7], 否, 是, 否, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)

Wave6第六層核對=([o7, u7], [u7], [o7], 否, 是, 否, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。

本題逐步詳解（AI 二次覆核）

Hausdorff空間中的緊緻集為閉集 · qf-mod43-compactclosed-nonhausdorff-boundary · candidate

84. Hausdorff空間中的緊緻集為閉集

ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-nonhausdorff-boundary；資料={"points":["o8","u8"],"topology":[["o8","u8"],["u8"],[]],"compact_subset":["u8"]}。請依 [points, compact_subset, complement, Hausdorff, compact, closed, boundary] 欄序回答。

查看答案與解釋

([o8, u8], [u8], [o8], 否, 是, 否, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)

Wave6第六層核對=([o8, u8], [u8], [o8], 否, 是, 否, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。

本題逐步詳解（AI 二次覆核）

Hausdorff空間中的緊緻集為閉集 · qf-mod43-compactclosed-nonhausdorff-boundary · candidate

85. Hausdorff空間中的緊緻集為閉集

ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-nonhausdorff-boundary；資料={"points":["o9","u9"],"topology":[["o9","u9"],["u9"],[]],"compact_subset":["u9"]}。請依 [points, compact_subset, complement, Hausdorff, compact, closed, boundary] 欄序回答。

查看答案與解釋

([o9, u9], [u9], [o9], 否, 是, 否, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)

Wave6第六層核對=([o9, u9], [u9], [o9], 否, 是, 否, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。

本題逐步詳解（AI 二次覆核）

Hausdorff空間中的緊緻集為閉集 · qf-mod43-compactclosed-nonhausdorff-boundary · candidate

86. Hausdorff空間中的緊緻集為閉集

ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-nonhausdorff-boundary；資料={"points":["o10","u10"],"topology":[["o10","u10"],["u10"],[]],"compact_subset":["u10"]}。請依 [points, compact_subset, complement, Hausdorff, compact, closed, boundary] 欄序回答。

查看答案與解釋

([o10, u10], [u10], [o10], 否, 是, 否, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)

Wave6第六層核對=([o10, u10], [u10], [o10], 否, 是, 否, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。

本題逐步詳解（AI 二次覆核）

Hausdorff空間中的緊緻集為閉集 · qf-mod43-compactclosed-nonhausdorff-boundary · candidate

87. Hausdorff空間中的緊緻集為閉集

ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-nonhausdorff-boundary；資料={"points":["o11","u11"],"topology":[["o11","u11"],["u11"],[]],"compact_subset":["u11"]}。請依 [points, compact_subset, complement, Hausdorff, compact, closed, boundary] 欄序回答。

查看答案與解釋

([o11, u11], [u11], [o11], 否, 是, 否, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)

Wave6第六層核對=([o11, u11], [u11], [o11], 否, 是, 否, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。

本題逐步詳解（AI 二次覆核）

Hausdorff空間中的緊緻集為閉集 · qf-mod43-compactclosed-nonhausdorff-boundary · candidate

88. Hausdorff空間中的緊緻集為閉集

ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-nonhausdorff-boundary；資料={"points":["o12","u12"],"topology":[["o12","u12"],["u12"],[]],"compact_subset":["u12"]}。請依 [points, compact_subset, complement, Hausdorff, compact, closed, boundary] 欄序回答。

查看答案與解釋

([o12, u12], [u12], [o12], 否, 是, 否, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)

Wave6第六層核對=([o12, u12], [u12], [o12], 否, 是, 否, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。

本題逐步詳解（AI 二次覆核）

Hausdorff空間中的緊緻集為閉集 · qf-mod43-compactclosed-nonhausdorff-boundary · candidate

89. Hausdorff空間中的緊緻集為閉集

ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-nonhausdorff-boundary；資料={"points":["o13","u13"],"topology":[["o13","u13"],["u13"],[]],"compact_subset":["u13"]}。請依 [points, compact_subset, complement, Hausdorff, compact, closed, boundary] 欄序回答。

查看答案與解釋

([o13, u13], [u13], [o13], 否, 是, 否, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)

Wave6第六層核對=([o13, u13], [u13], [o13], 否, 是, 否, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。

本題逐步詳解（AI 二次覆核）

Hausdorff空間中的緊緻集為閉集 · qf-mod43-compactclosed-nonhausdorff-boundary · candidate

90. Hausdorff空間中的緊緻集為閉集

ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-nonhausdorff-boundary；資料={"points":["o14","u14"],"topology":[["o14","u14"],["u14"],[]],"compact_subset":["u14"]}。請依 [points, compact_subset, complement, Hausdorff, compact, closed, boundary] 欄序回答。

查看答案與解釋

([o14, u14], [u14], [o14], 否, 是, 否, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)

Wave6第六層核對=([o14, u14], [u14], [o14], 否, 是, 否, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。

本題逐步詳解（AI 二次覆核）

Hausdorff空間中的緊緻集為閉集 · qf-mod43-compactclosed-nonhausdorff-boundary · candidate

91. Hausdorff空間中的緊緻集為閉集

ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-nonhausdorff-boundary；資料={"points":["o15","u15"],"topology":[["o15","u15"],["u15"],[]],"compact_subset":["u15"]}。請依 [points, compact_subset, complement, Hausdorff, compact, closed, boundary] 欄序回答。

查看答案與解釋

([o15, u15], [u15], [o15], 否, 是, 否, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)

Wave6第六層核對=([o15, u15], [u15], [o15], 否, 是, 否, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。

本題逐步詳解（AI 二次覆核）

Hausdorff空間中的緊緻集為閉集 · qf-mod43-compactclosed-nonhausdorff-boundary · candidate

92. Hausdorff空間中的緊緻集為閉集

ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-nonhausdorff-boundary；資料={"points":["o16","u16"],"topology":[["o16","u16"],["u16"],[]],"compact_subset":["u16"]}。請依 [points, compact_subset, complement, Hausdorff, compact, closed, boundary] 欄序回答。

查看答案與解釋

([o16, u16], [u16], [o16], 否, 是, 否, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)

Wave6第六層核對=([o16, u16], [u16], [o16], 否, 是, 否, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。

本題逐步詳解（AI 二次覆核）

Hausdorff空間中的緊緻集為閉集 · qf-mod43-compactclosed-nonhausdorff-boundary · candidate

93. Hausdorff空間中的緊緻集為閉集

ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-nonhausdorff-boundary；資料={"points":["o17","u17"],"topology":[["o17","u17"],["u17"],[]],"compact_subset":["u17"]}。請依 [points, compact_subset, complement, Hausdorff, compact, closed, boundary] 欄序回答。

查看答案與解釋

([o17, u17], [u17], [o17], 否, 是, 否, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)

Wave6第六層核對=([o17, u17], [u17], [o17], 否, 是, 否, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。

本題逐步詳解（AI 二次覆核）

Hausdorff空間中的緊緻集為閉集 · qf-mod43-compactclosed-nonhausdorff-boundary · candidate

94. Hausdorff空間中的緊緻集為閉集

ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-nonhausdorff-boundary；資料={"points":["o18","u18"],"topology":[["o18","u18"],["u18"],[]],"compact_subset":["u18"]}。請依 [points, compact_subset, complement, Hausdorff, compact, closed, boundary] 欄序回答。

查看答案與解釋

([o18, u18], [u18], [o18], 否, 是, 否, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)

Wave6第六層核對=([o18, u18], [u18], [o18], 否, 是, 否, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。

本題逐步詳解（AI 二次覆核）

Hausdorff空間中的緊緻集為閉集 · qf-mod43-compactclosed-nonhausdorff-boundary · candidate

95. Hausdorff空間中的緊緻集為閉集

ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-nonhausdorff-boundary；資料={"points":["o19","u19"],"topology":[["o19","u19"],["u19"],[]],"compact_subset":["u19"]}。請依 [points, compact_subset, complement, Hausdorff, compact, closed, boundary] 欄序回答。

查看答案與解釋

([o19, u19], [u19], [o19], 否, 是, 否, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)

Wave6第六層核對=([o19, u19], [u19], [o19], 否, 是, 否, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。

本題逐步詳解（AI 二次覆核）

Hausdorff空間中的緊緻集為閉集 · qf-mod43-compactclosed-nonhausdorff-boundary · candidate

96. Hausdorff空間中的緊緻集為閉集

ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-nonhausdorff-boundary；資料={"points":["o20","u20"],"topology":[["o20","u20"],["u20"],[]],"compact_subset":["u20"]}。請依 [points, compact_subset, complement, Hausdorff, compact, closed, boundary] 欄序回答。

查看答案與解釋

([o20, u20], [u20], [o20], 否, 是, 否, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)

Wave6第六層核對=([o20, u20], [u20], [o20], 否, 是, 否, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。

本題逐步詳解（AI 二次覆核）

Hausdorff空間中的緊緻集為閉集 · qf-mod43-compactclosed-nonhausdorff-boundary · candidate

97. Hausdorff空間中的緊緻集為閉集

ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-nonhausdorff-boundary；資料={"points":["o21","u21"],"topology":[["o21","u21"],["u21"],[]],"compact_subset":["u21"]}。請依 [points, compact_subset, complement, Hausdorff, compact, closed, boundary] 欄序回答。

查看答案與解釋

([o21, u21], [u21], [o21], 否, 是, 否, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)

Wave6第六層核對=([o21, u21], [u21], [o21], 否, 是, 否, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。

本題逐步詳解（AI 二次覆核）

Hausdorff空間中的緊緻集為閉集 · qf-mod43-compactclosed-nonhausdorff-boundary · candidate

98. Hausdorff空間中的緊緻集為閉集

ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-nonhausdorff-boundary；資料={"points":["o22","u22"],"topology":[["o22","u22"],["u22"],[]],"compact_subset":["u22"]}。請依 [points, compact_subset, complement, Hausdorff, compact, closed, boundary] 欄序回答。

查看答案與解釋

([o22, u22], [u22], [o22], 否, 是, 否, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)

Wave6第六層核對=([o22, u22], [u22], [o22], 否, 是, 否, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。

本題逐步詳解（AI 二次覆核）

Hausdorff空間中的緊緻集為閉集 · qf-mod43-compactclosed-nonhausdorff-boundary · candidate

99. Hausdorff空間中的緊緻集為閉集

ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-nonhausdorff-boundary；資料={"points":["o23","u23"],"topology":[["o23","u23"],["u23"],[]],"compact_subset":["u23"]}。請依 [points, compact_subset, complement, Hausdorff, compact, closed, boundary] 欄序回答。

查看答案與解釋

([o23, u23], [u23], [o23], 否, 是, 否, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)

Wave6第六層核對=([o23, u23], [u23], [o23], 否, 是, 否, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。

本題逐步詳解（AI 二次覆核）

Hausdorff空間中的緊緻集為閉集 · qf-mod43-compactclosed-nonhausdorff-boundary · candidate

100. Hausdorff空間中的緊緻集為閉集

ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-nonhausdorff-boundary；資料={"points":["o24","u24"],"topology":[["o24","u24"],["u24"],[]],"compact_subset":["u24"]}。請依 [points, compact_subset, complement, Hausdorff, compact, closed, boundary] 欄序回答。

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([o24, u24], [u24], [o24], 否, 是, 否, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)

Wave6第六層核對=([o24, u24], [u24], [o24], 否, 是, 否, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。

本題逐步詳解（AI 二次覆核）

Hausdorff空間中的緊緻集為閉集 · qf-mod43-compactclosed-nonhausdorff-boundary · candidate

原始題目 JSONL · 驗收報告