# Complete Point-Pair Separators, Vacuous Cases and Sample Nonconclusions

This batch has passed Schema, deduplication, source-snapshot, and independent-answer recomputation checks; it is still not marked as human-reviewed.

This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.

batch-20261014-advance-20261111-r03-hausdorff-space-foundations · 2026-10-14 · 100 questions

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

Field order uses a source-bound presentation supplement; canonical JSONL and mathematical answers are unchanged.

[Field-order and source-binding data](https://ocme.evemisslab.com/data/i18n/question-output-protocols.json)



## 1. A Pair Separator Must Use Two Actual Disjoint Open Neighborhoods

Use finite unique labeled points=[a0, b0, c0] and the complete canonical topology=[[a0, b0, c0], [a0, b0], [a0, c0], [a0], [b0, c0], [b0], [c0], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Test x=a0, y=b0. Give (x, y, open_neighborhood_witness, separated, witness_disjoint).

View answer and explanation(a0, b0, [[a0, c0], [b0]], Yes, Yes)

Search every open pair in source canonical order and retain the first valid witness, or null if none exists. Report separation and actual disjointness of the witness separately; a null witness supplies neither flag. The exact result, retaining the requested order, is (a0, b0, [[a0, c0], [b0]], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r03-hausdorff-space-0001/)

Hausdorff space · qf-mod42-hausdorff-pair-separator · candidate





## 2. A Pair Separator Must Use Two Actual Disjoint Open Neighborhoods

Use finite unique labeled points=[a1, b1, c1] and the complete canonical topology=[[a1, b1, c1], [a1, b1], [a1, c1], [a1], [b1, c1], [b1], [c1], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Test x=a1, y=b1. Give (x, y, open_neighborhood_witness, separated, witness_disjoint).

View answer and explanation(a1, b1, [[a1, c1], [b1]], Yes, Yes)

Search every open pair in source canonical order and retain the first valid witness, or null if none exists. Report separation and actual disjointness of the witness separately; a null witness supplies neither flag. The exact result, retaining the requested order, is (a1, b1, [[a1, c1], [b1]], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r03-hausdorff-space-0002/)

Hausdorff space · qf-mod42-hausdorff-pair-separator · candidate





## 3. A Pair Separator Must Use Two Actual Disjoint Open Neighborhoods

Use finite unique labeled points=[a2, b2, c2] and the complete canonical topology=[[a2, b2, c2], [a2, b2], [a2, c2], [a2], [b2, c2], [b2], [c2], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Test x=a2, y=b2. Give (x, y, open_neighborhood_witness, separated, witness_disjoint).

View answer and explanation(a2, b2, [[a2, c2], [b2]], Yes, Yes)

Search every open pair in source canonical order and retain the first valid witness, or null if none exists. Report separation and actual disjointness of the witness separately; a null witness supplies neither flag. The exact result, retaining the requested order, is (a2, b2, [[a2, c2], [b2]], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r03-hausdorff-space-0003/)

Hausdorff space · qf-mod42-hausdorff-pair-separator · candidate





## 4. A Pair Separator Must Use Two Actual Disjoint Open Neighborhoods

Use finite unique labeled points=[a3, b3, c3] and the complete canonical topology=[[a3, b3, c3], [a3, b3], [a3, c3], [a3], [b3, c3], [b3], [c3], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Test x=a3, y=b3. Give (x, y, open_neighborhood_witness, separated, witness_disjoint).

View answer and explanation(a3, b3, [[a3, c3], [b3]], Yes, Yes)

Search every open pair in source canonical order and retain the first valid witness, or null if none exists. Report separation and actual disjointness of the witness separately; a null witness supplies neither flag. The exact result, retaining the requested order, is (a3, b3, [[a3, c3], [b3]], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r03-hausdorff-space-0004/)

Hausdorff space · qf-mod42-hausdorff-pair-separator · candidate





## 5. A Pair Separator Must Use Two Actual Disjoint Open Neighborhoods

Use finite unique labeled points=[a4, b4, c4] and the complete canonical topology=[[a4, b4, c4], [a4, b4], [a4, c4], [a4], [b4, c4], [b4], [c4], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Test x=a4, y=b4. Give (x, y, open_neighborhood_witness, separated, witness_disjoint).

View answer and explanation(a4, b4, [[a4, c4], [b4]], Yes, Yes)

Search every open pair in source canonical order and retain the first valid witness, or null if none exists. Report separation and actual disjointness of the witness separately; a null witness supplies neither flag. The exact result, retaining the requested order, is (a4, b4, [[a4, c4], [b4]], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r03-hausdorff-space-0005/)

Hausdorff space · qf-mod42-hausdorff-pair-separator · candidate





## 6. A Pair Separator Must Use Two Actual Disjoint Open Neighborhoods

Use finite unique labeled points=[a5, b5, c5] and the complete canonical topology=[[a5, b5, c5], [a5, b5], [a5, c5], [a5], [b5, c5], [b5], [c5], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Test x=a5, y=b5. Give (x, y, open_neighborhood_witness, separated, witness_disjoint).

View answer and explanation(a5, b5, [[a5, c5], [b5]], Yes, Yes)

Search every open pair in source canonical order and retain the first valid witness, or null if none exists. Report separation and actual disjointness of the witness separately; a null witness supplies neither flag. The exact result, retaining the requested order, is (a5, b5, [[a5, c5], [b5]], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r03-hausdorff-space-0006/)

Hausdorff space · qf-mod42-hausdorff-pair-separator · candidate





## 7. A Pair Separator Must Use Two Actual Disjoint Open Neighborhoods

Use finite unique labeled points=[a6, b6, c6] and the complete canonical topology=[[a6, b6, c6], [a6, b6], [a6, c6], [a6], [b6, c6], [b6], [c6], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Test x=a6, y=b6. Give (x, y, open_neighborhood_witness, separated, witness_disjoint).

View answer and explanation(a6, b6, [[a6, c6], [b6]], Yes, Yes)

Search every open pair in source canonical order and retain the first valid witness, or null if none exists. Report separation and actual disjointness of the witness separately; a null witness supplies neither flag. The exact result, retaining the requested order, is (a6, b6, [[a6, c6], [b6]], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r03-hausdorff-space-0007/)

Hausdorff space · qf-mod42-hausdorff-pair-separator · candidate





## 8. A Pair Separator Must Use Two Actual Disjoint Open Neighborhoods

Use finite unique labeled points=[a7, b7, c7] and the complete canonical topology=[[a7, b7, c7], [a7, b7], [a7, c7], [a7], [b7, c7], [b7], [c7], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Test x=a7, y=b7. Give (x, y, open_neighborhood_witness, separated, witness_disjoint).

View answer and explanation(a7, b7, [[a7, c7], [b7]], Yes, Yes)

Search every open pair in source canonical order and retain the first valid witness, or null if none exists. Report separation and actual disjointness of the witness separately; a null witness supplies neither flag. The exact result, retaining the requested order, is (a7, b7, [[a7, c7], [b7]], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r03-hausdorff-space-0008/)

Hausdorff space · qf-mod42-hausdorff-pair-separator · candidate





## 9. A Pair Separator Must Use Two Actual Disjoint Open Neighborhoods

Use finite unique labeled points=[a8, b8, c8] and the complete canonical topology=[[a8, b8, c8], [a8, b8], [a8, c8], [a8], [b8, c8], [b8], [c8], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Test x=a8, y=b8. Give (x, y, open_neighborhood_witness, separated, witness_disjoint).

View answer and explanation(a8, b8, [[a8, c8], [b8]], Yes, Yes)

Search every open pair in source canonical order and retain the first valid witness, or null if none exists. Report separation and actual disjointness of the witness separately; a null witness supplies neither flag. The exact result, retaining the requested order, is (a8, b8, [[a8, c8], [b8]], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r03-hausdorff-space-0009/)

Hausdorff space · qf-mod42-hausdorff-pair-separator · candidate





## 10. A Pair Separator Must Use Two Actual Disjoint Open Neighborhoods

Use finite unique labeled points=[a9, b9, c9] and the complete canonical topology=[[a9, b9, c9], [a9, b9], [a9, c9], [a9], [b9, c9], [b9], [c9], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Test x=a9, y=b9. Give (x, y, open_neighborhood_witness, separated, witness_disjoint).

View answer and explanation(a9, b9, [[a9, c9], [b9]], Yes, Yes)

Search every open pair in source canonical order and retain the first valid witness, or null if none exists. Report separation and actual disjointness of the witness separately; a null witness supplies neither flag. The exact result, retaining the requested order, is (a9, b9, [[a9, c9], [b9]], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r03-hausdorff-space-0010/)

Hausdorff space · qf-mod42-hausdorff-pair-separator · candidate





## 11. A Pair Separator Must Use Two Actual Disjoint Open Neighborhoods

Use finite unique labeled points=[a10, b10, c10] and the complete canonical topology=[[a10, b10, c10], [a10, b10], [a10, c10], [a10], [b10, c10], [b10], [c10], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Test x=a10, y=b10. Give (x, y, open_neighborhood_witness, separated, witness_disjoint).

View answer and explanation(a10, b10, [[a10, c10], [b10]], Yes, Yes)

Search every open pair in source canonical order and retain the first valid witness, or null if none exists. Report separation and actual disjointness of the witness separately; a null witness supplies neither flag. The exact result, retaining the requested order, is (a10, b10, [[a10, c10], [b10]], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r03-hausdorff-space-0011/)

Hausdorff space · qf-mod42-hausdorff-pair-separator · candidate





## 12. A Pair Separator Must Use Two Actual Disjoint Open Neighborhoods

Use finite unique labeled points=[a11, b11, c11] and the complete canonical topology=[[a11, b11, c11], [a11, b11], [a11, c11], [a11], [b11, c11], [b11], [c11], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Test x=a11, y=b11. Give (x, y, open_neighborhood_witness, separated, witness_disjoint).

View answer and explanation(a11, b11, [[a11, c11], [b11]], Yes, Yes)

Search every open pair in source canonical order and retain the first valid witness, or null if none exists. Report separation and actual disjointness of the witness separately; a null witness supplies neither flag. The exact result, retaining the requested order, is (a11, b11, [[a11, c11], [b11]], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r03-hausdorff-space-0012/)

Hausdorff space · qf-mod42-hausdorff-pair-separator · candidate





## 13. A Pair Separator Must Use Two Actual Disjoint Open Neighborhoods

Use finite unique labeled points=[a12, b12, c12] and the complete canonical topology=[[a12, b12, c12], [a12, b12], [a12, c12], [a12], [b12, c12], [b12], [c12], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Test x=a12, y=b12. Give (x, y, open_neighborhood_witness, separated, witness_disjoint).

View answer and explanation(a12, b12, [[a12, c12], [b12]], Yes, Yes)

Search every open pair in source canonical order and retain the first valid witness, or null if none exists. Report separation and actual disjointness of the witness separately; a null witness supplies neither flag. The exact result, retaining the requested order, is (a12, b12, [[a12, c12], [b12]], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r03-hausdorff-space-0013/)

Hausdorff space · qf-mod42-hausdorff-pair-separator · candidate





## 14. A Pair Separator Must Use Two Actual Disjoint Open Neighborhoods

Use finite unique labeled points=[a13, b13, c13] and the complete canonical topology=[[a13, b13, c13], [a13, b13], [a13, c13], [a13], [b13, c13], [b13], [c13], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Test x=a13, y=b13. Give (x, y, open_neighborhood_witness, separated, witness_disjoint).

View answer and explanation(a13, b13, [[a13, c13], [b13]], Yes, Yes)

Search every open pair in source canonical order and retain the first valid witness, or null if none exists. Report separation and actual disjointness of the witness separately; a null witness supplies neither flag. The exact result, retaining the requested order, is (a13, b13, [[a13, c13], [b13]], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r03-hausdorff-space-0014/)

Hausdorff space · qf-mod42-hausdorff-pair-separator · candidate





## 15. A Pair Separator Must Use Two Actual Disjoint Open Neighborhoods

Use finite unique labeled points=[a14, b14, c14] and the complete canonical topology=[[a14, b14, c14], [a14, b14], [a14, c14], [a14], [b14, c14], [b14], [c14], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Test x=a14, y=b14. Give (x, y, open_neighborhood_witness, separated, witness_disjoint).

View answer and explanation(a14, b14, [[a14, c14], [b14]], Yes, Yes)

Search every open pair in source canonical order and retain the first valid witness, or null if none exists. Report separation and actual disjointness of the witness separately; a null witness supplies neither flag. The exact result, retaining the requested order, is (a14, b14, [[a14, c14], [b14]], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r03-hausdorff-space-0015/)

Hausdorff space · qf-mod42-hausdorff-pair-separator · candidate





## 16. A Pair Separator Must Use Two Actual Disjoint Open Neighborhoods

Use finite unique labeled points=[a15, b15, c15] and the complete canonical topology=[[a15, b15, c15], [a15, b15], [a15, c15], [a15], [b15, c15], [b15], [c15], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Test x=a15, y=b15. Give (x, y, open_neighborhood_witness, separated, witness_disjoint).

View answer and explanation(a15, b15, [[a15, c15], [b15]], Yes, Yes)

Search every open pair in source canonical order and retain the first valid witness, or null if none exists. Report separation and actual disjointness of the witness separately; a null witness supplies neither flag. The exact result, retaining the requested order, is (a15, b15, [[a15, c15], [b15]], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r03-hausdorff-space-0016/)

Hausdorff space · qf-mod42-hausdorff-pair-separator · candidate





## 17. A Pair Separator Must Use Two Actual Disjoint Open Neighborhoods

Use finite unique labeled points=[a16, b16, c16] and the complete canonical topology=[[a16, b16, c16], [a16, b16], [a16, c16], [a16], [b16, c16], [b16], [c16], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Test x=a16, y=b16. Give (x, y, open_neighborhood_witness, separated, witness_disjoint).

View answer and explanation(a16, b16, [[a16, c16], [b16]], Yes, Yes)

Search every open pair in source canonical order and retain the first valid witness, or null if none exists. Report separation and actual disjointness of the witness separately; a null witness supplies neither flag. The exact result, retaining the requested order, is (a16, b16, [[a16, c16], [b16]], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r03-hausdorff-space-0017/)

Hausdorff space · qf-mod42-hausdorff-pair-separator · candidate





## 18. A Pair Separator Must Use Two Actual Disjoint Open Neighborhoods

Use finite unique labeled points=[a17, b17, c17] and the complete canonical topology=[[a17, b17, c17], [a17, b17], [a17, c17], [a17], [b17, c17], [b17], [c17], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Test x=a17, y=b17. Give (x, y, open_neighborhood_witness, separated, witness_disjoint).

View answer and explanation(a17, b17, [[a17, c17], [b17]], Yes, Yes)

Search every open pair in source canonical order and retain the first valid witness, or null if none exists. Report separation and actual disjointness of the witness separately; a null witness supplies neither flag. The exact result, retaining the requested order, is (a17, b17, [[a17, c17], [b17]], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r03-hausdorff-space-0018/)

Hausdorff space · qf-mod42-hausdorff-pair-separator · candidate





## 19. A Pair Separator Must Use Two Actual Disjoint Open Neighborhoods

Use finite unique labeled points=[a18, b18, c18] and the complete canonical topology=[[a18, b18, c18], [a18, b18], [a18, c18], [a18], [b18, c18], [b18], [c18], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Test x=a18, y=b18. Give (x, y, open_neighborhood_witness, separated, witness_disjoint).

View answer and explanation(a18, b18, [[a18, c18], [b18]], Yes, Yes)

Search every open pair in source canonical order and retain the first valid witness, or null if none exists. Report separation and actual disjointness of the witness separately; a null witness supplies neither flag. The exact result, retaining the requested order, is (a18, b18, [[a18, c18], [b18]], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r03-hausdorff-space-0019/)

Hausdorff space · qf-mod42-hausdorff-pair-separator · candidate





## 20. A Pair Separator Must Use Two Actual Disjoint Open Neighborhoods

Use finite unique labeled points=[a19, b19, c19] and the complete canonical topology=[[a19, b19, c19], [a19, b19], [a19, c19], [a19], [b19, c19], [b19], [c19], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Test x=a19, y=b19. Give (x, y, open_neighborhood_witness, separated, witness_disjoint).

View answer and explanation(a19, b19, [[a19, c19], [b19]], Yes, Yes)

Search every open pair in source canonical order and retain the first valid witness, or null if none exists. Report separation and actual disjointness of the witness separately; a null witness supplies neither flag. The exact result, retaining the requested order, is (a19, b19, [[a19, c19], [b19]], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r03-hausdorff-space-0020/)

Hausdorff space · qf-mod42-hausdorff-pair-separator · candidate





## 21. A Pair Separator Must Use Two Actual Disjoint Open Neighborhoods

Use finite unique labeled points=[a20, b20, c20] and the complete canonical topology=[[a20, b20, c20], [a20, b20], [a20, c20], [a20], [b20, c20], [b20], [c20], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Test x=a20, y=b20. Give (x, y, open_neighborhood_witness, separated, witness_disjoint).

View answer and explanation(a20, b20, [[a20, c20], [b20]], Yes, Yes)

Search every open pair in source canonical order and retain the first valid witness, or null if none exists. Report separation and actual disjointness of the witness separately; a null witness supplies neither flag. The exact result, retaining the requested order, is (a20, b20, [[a20, c20], [b20]], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r03-hausdorff-space-0021/)

Hausdorff space · qf-mod42-hausdorff-pair-separator · candidate





## 22. A Pair Separator Must Use Two Actual Disjoint Open Neighborhoods

Use finite unique labeled points=[a21, b21, c21] and the complete canonical topology=[[a21, b21, c21], [a21, b21], [a21, c21], [a21], [b21, c21], [b21], [c21], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Test x=a21, y=b21. Give (x, y, open_neighborhood_witness, separated, witness_disjoint).

View answer and explanation(a21, b21, [[a21, c21], [b21]], Yes, Yes)

Search every open pair in source canonical order and retain the first valid witness, or null if none exists. Report separation and actual disjointness of the witness separately; a null witness supplies neither flag. The exact result, retaining the requested order, is (a21, b21, [[a21, c21], [b21]], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r03-hausdorff-space-0022/)

Hausdorff space · qf-mod42-hausdorff-pair-separator · candidate





## 23. A Pair Separator Must Use Two Actual Disjoint Open Neighborhoods

Use finite unique labeled points=[a22, b22, c22] and the complete canonical topology=[[a22, b22, c22], [a22, b22], [a22, c22], [a22], [b22, c22], [b22], [c22], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Test x=a22, y=b22. Give (x, y, open_neighborhood_witness, separated, witness_disjoint).

View answer and explanation(a22, b22, [[a22, c22], [b22]], Yes, Yes)

Search every open pair in source canonical order and retain the first valid witness, or null if none exists. Report separation and actual disjointness of the witness separately; a null witness supplies neither flag. The exact result, retaining the requested order, is (a22, b22, [[a22, c22], [b22]], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r03-hausdorff-space-0023/)

Hausdorff space · qf-mod42-hausdorff-pair-separator · candidate





## 24. A Pair Separator Must Use Two Actual Disjoint Open Neighborhoods

Use finite unique labeled points=[a23, b23, c23] and the complete canonical topology=[[a23, b23, c23], [a23, b23], [a23, c23], [a23], [b23, c23], [b23], [c23], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Test x=a23, y=b23. Give (x, y, open_neighborhood_witness, separated, witness_disjoint).

View answer and explanation(a23, b23, [[a23, c23], [b23]], Yes, Yes)

Search every open pair in source canonical order and retain the first valid witness, or null if none exists. Report separation and actual disjointness of the witness separately; a null witness supplies neither flag. The exact result, retaining the requested order, is (a23, b23, [[a23, c23], [b23]], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r03-hausdorff-space-0024/)

Hausdorff space · qf-mod42-hausdorff-pair-separator · candidate





## 25. A Pair Separator Must Use Two Actual Disjoint Open Neighborhoods

Use finite unique labeled points=[a24, b24, c24] and the complete canonical topology=[[a24, b24, c24], [a24, b24], [a24, c24], [a24], [b24, c24], [b24], [c24], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Test x=a24, y=b24. Give (x, y, open_neighborhood_witness, separated, witness_disjoint).

View answer and explanation(a24, b24, [[a24, c24], [b24]], Yes, Yes)

Search every open pair in source canonical order and retain the first valid witness, or null if none exists. Report separation and actual disjointness of the witness separately; a null witness supplies neither flag. The exact result, retaining the requested order, is (a24, b24, [[a24, c24], [b24]], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r03-hausdorff-space-0025/)

Hausdorff space · qf-mod42-hausdorff-pair-separator · candidate





## 26. All Distinct Point Pairs Must Be Separated in the Complete Finite Topology

Use finite unique labeled points=[a0, b0, c0] and the complete canonical topology=[[a0, b0, c0], [a0, b0], [a0, c0], [a0], [b0, c0], [b0], [c0], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Give (points, pair_witness_rows, Hausdorff, certificate).

View answer and explanation([a0, b0, c0], [[a0, b0, [[a0, c0], [b0]]], [a0, c0, [[a0, b0], [c0]]], [b0, c0, [[a0, b0], [c0]]]], Yes, all_distinct_pairs_exhaustively_separated)

Enumerate every unordered pair in original point order and its first source-order witness. A single failed pair disproves Hausdorffness. The empty/singleton cases have no distinct pair and are vacuously Hausdorff. Finite Hausdorff implies discrete, but that corollary must not be extrapolated to infinite spaces. The exact result, retaining the requested order, is ([a0, b0, c0], [[a0, b0, [[a0, c0], [b0]]], [a0, c0, [[a0, b0], [c0]]], [b0, c0, [[a0, b0], [c0]]]], Yes, all_distinct_pairs_exhaustively_separated).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r03-hausdorff-space-0026/)

Hausdorff space · qf-mod42-hausdorff-full-audit · candidate





## 27. All Distinct Point Pairs Must Be Separated in the Complete Finite Topology

Use finite unique labeled points=[a1, b1, c1] and the complete canonical topology=[[a1, b1, c1], [a1, b1], [a1, c1], [a1], [b1, c1], [b1], [c1], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Give (points, pair_witness_rows, Hausdorff, certificate).

View answer and explanation([a1, b1, c1], [[a1, b1, [[a1, c1], [b1]]], [a1, c1, [[a1, b1], [c1]]], [b1, c1, [[a1, b1], [c1]]]], Yes, all_distinct_pairs_exhaustively_separated)

Enumerate every unordered pair in original point order and its first source-order witness. A single failed pair disproves Hausdorffness. The empty/singleton cases have no distinct pair and are vacuously Hausdorff. Finite Hausdorff implies discrete, but that corollary must not be extrapolated to infinite spaces. The exact result, retaining the requested order, is ([a1, b1, c1], [[a1, b1, [[a1, c1], [b1]]], [a1, c1, [[a1, b1], [c1]]], [b1, c1, [[a1, b1], [c1]]]], Yes, all_distinct_pairs_exhaustively_separated).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r03-hausdorff-space-0027/)

Hausdorff space · qf-mod42-hausdorff-full-audit · candidate





## 28. All Distinct Point Pairs Must Be Separated in the Complete Finite Topology

Use finite unique labeled points=[a2, b2, c2] and the complete canonical topology=[[a2, b2, c2], [a2, b2], [a2, c2], [a2], [b2, c2], [b2], [c2], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Give (points, pair_witness_rows, Hausdorff, certificate).

View answer and explanation([a2, b2, c2], [[a2, b2, [[a2, c2], [b2]]], [a2, c2, [[a2, b2], [c2]]], [b2, c2, [[a2, b2], [c2]]]], Yes, all_distinct_pairs_exhaustively_separated)

Enumerate every unordered pair in original point order and its first source-order witness. A single failed pair disproves Hausdorffness. The empty/singleton cases have no distinct pair and are vacuously Hausdorff. Finite Hausdorff implies discrete, but that corollary must not be extrapolated to infinite spaces. The exact result, retaining the requested order, is ([a2, b2, c2], [[a2, b2, [[a2, c2], [b2]]], [a2, c2, [[a2, b2], [c2]]], [b2, c2, [[a2, b2], [c2]]]], Yes, all_distinct_pairs_exhaustively_separated).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r03-hausdorff-space-0028/)

Hausdorff space · qf-mod42-hausdorff-full-audit · candidate





## 29. All Distinct Point Pairs Must Be Separated in the Complete Finite Topology

Use finite unique labeled points=[a3, b3, c3] and the complete canonical topology=[[a3, b3, c3], [a3, b3], [a3, c3], [a3], [b3, c3], [b3], [c3], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Give (points, pair_witness_rows, Hausdorff, certificate).

View answer and explanation([a3, b3, c3], [[a3, b3, [[a3, c3], [b3]]], [a3, c3, [[a3, b3], [c3]]], [b3, c3, [[a3, b3], [c3]]]], Yes, all_distinct_pairs_exhaustively_separated)

Enumerate every unordered pair in original point order and its first source-order witness. A single failed pair disproves Hausdorffness. The empty/singleton cases have no distinct pair and are vacuously Hausdorff. Finite Hausdorff implies discrete, but that corollary must not be extrapolated to infinite spaces. The exact result, retaining the requested order, is ([a3, b3, c3], [[a3, b3, [[a3, c3], [b3]]], [a3, c3, [[a3, b3], [c3]]], [b3, c3, [[a3, b3], [c3]]]], Yes, all_distinct_pairs_exhaustively_separated).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r03-hausdorff-space-0029/)

Hausdorff space · qf-mod42-hausdorff-full-audit · candidate





## 30. All Distinct Point Pairs Must Be Separated in the Complete Finite Topology

Use finite unique labeled points=[a4, b4, c4] and the complete canonical topology=[[a4, b4, c4], [a4, b4], [a4, c4], [a4], [b4, c4], [b4], [c4], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Give (points, pair_witness_rows, Hausdorff, certificate).

View answer and explanation([a4, b4, c4], [[a4, b4, [[a4, c4], [b4]]], [a4, c4, [[a4, b4], [c4]]], [b4, c4, [[a4, b4], [c4]]]], Yes, all_distinct_pairs_exhaustively_separated)

Enumerate every unordered pair in original point order and its first source-order witness. A single failed pair disproves Hausdorffness. The empty/singleton cases have no distinct pair and are vacuously Hausdorff. Finite Hausdorff implies discrete, but that corollary must not be extrapolated to infinite spaces. The exact result, retaining the requested order, is ([a4, b4, c4], [[a4, b4, [[a4, c4], [b4]]], [a4, c4, [[a4, b4], [c4]]], [b4, c4, [[a4, b4], [c4]]]], Yes, all_distinct_pairs_exhaustively_separated).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r03-hausdorff-space-0030/)

Hausdorff space · qf-mod42-hausdorff-full-audit · candidate





## 31. All Distinct Point Pairs Must Be Separated in the Complete Finite Topology

Use finite unique labeled points=[a5, b5, c5] and the complete canonical topology=[[a5, b5, c5], [a5, b5], [a5, c5], [a5], [b5, c5], [b5], [c5], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Give (points, pair_witness_rows, Hausdorff, certificate).

View answer and explanation([a5, b5, c5], [[a5, b5, [[a5, c5], [b5]]], [a5, c5, [[a5, b5], [c5]]], [b5, c5, [[a5, b5], [c5]]]], Yes, all_distinct_pairs_exhaustively_separated)

Enumerate every unordered pair in original point order and its first source-order witness. A single failed pair disproves Hausdorffness. The empty/singleton cases have no distinct pair and are vacuously Hausdorff. Finite Hausdorff implies discrete, but that corollary must not be extrapolated to infinite spaces. The exact result, retaining the requested order, is ([a5, b5, c5], [[a5, b5, [[a5, c5], [b5]]], [a5, c5, [[a5, b5], [c5]]], [b5, c5, [[a5, b5], [c5]]]], Yes, all_distinct_pairs_exhaustively_separated).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r03-hausdorff-space-0031/)

Hausdorff space · qf-mod42-hausdorff-full-audit · candidate





## 32. All Distinct Point Pairs Must Be Separated in the Complete Finite Topology

Use finite unique labeled points=[a6, b6, c6] and the complete canonical topology=[[a6, b6, c6], [a6, b6], [a6, c6], [a6], [b6, c6], [b6], [c6], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Give (points, pair_witness_rows, Hausdorff, certificate).

View answer and explanation([a6, b6, c6], [[a6, b6, [[a6, c6], [b6]]], [a6, c6, [[a6, b6], [c6]]], [b6, c6, [[a6, b6], [c6]]]], Yes, all_distinct_pairs_exhaustively_separated)

Enumerate every unordered pair in original point order and its first source-order witness. A single failed pair disproves Hausdorffness. The empty/singleton cases have no distinct pair and are vacuously Hausdorff. Finite Hausdorff implies discrete, but that corollary must not be extrapolated to infinite spaces. The exact result, retaining the requested order, is ([a6, b6, c6], [[a6, b6, [[a6, c6], [b6]]], [a6, c6, [[a6, b6], [c6]]], [b6, c6, [[a6, b6], [c6]]]], Yes, all_distinct_pairs_exhaustively_separated).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r03-hausdorff-space-0032/)

Hausdorff space · qf-mod42-hausdorff-full-audit · candidate





## 33. All Distinct Point Pairs Must Be Separated in the Complete Finite Topology

Use finite unique labeled points=[a7, b7, c7] and the complete canonical topology=[[a7, b7, c7], [a7, b7], [a7, c7], [a7], [b7, c7], [b7], [c7], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Give (points, pair_witness_rows, Hausdorff, certificate).

View answer and explanation([a7, b7, c7], [[a7, b7, [[a7, c7], [b7]]], [a7, c7, [[a7, b7], [c7]]], [b7, c7, [[a7, b7], [c7]]]], Yes, all_distinct_pairs_exhaustively_separated)

Enumerate every unordered pair in original point order and its first source-order witness. A single failed pair disproves Hausdorffness. The empty/singleton cases have no distinct pair and are vacuously Hausdorff. Finite Hausdorff implies discrete, but that corollary must not be extrapolated to infinite spaces. The exact result, retaining the requested order, is ([a7, b7, c7], [[a7, b7, [[a7, c7], [b7]]], [a7, c7, [[a7, b7], [c7]]], [b7, c7, [[a7, b7], [c7]]]], Yes, all_distinct_pairs_exhaustively_separated).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r03-hausdorff-space-0033/)

Hausdorff space · qf-mod42-hausdorff-full-audit · candidate





## 34. All Distinct Point Pairs Must Be Separated in the Complete Finite Topology

Use finite unique labeled points=[a8, b8, c8] and the complete canonical topology=[[a8, b8, c8], [a8, b8], [a8, c8], [a8], [b8, c8], [b8], [c8], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Give (points, pair_witness_rows, Hausdorff, certificate).

View answer and explanation([a8, b8, c8], [[a8, b8, [[a8, c8], [b8]]], [a8, c8, [[a8, b8], [c8]]], [b8, c8, [[a8, b8], [c8]]]], Yes, all_distinct_pairs_exhaustively_separated)

Enumerate every unordered pair in original point order and its first source-order witness. A single failed pair disproves Hausdorffness. The empty/singleton cases have no distinct pair and are vacuously Hausdorff. Finite Hausdorff implies discrete, but that corollary must not be extrapolated to infinite spaces. The exact result, retaining the requested order, is ([a8, b8, c8], [[a8, b8, [[a8, c8], [b8]]], [a8, c8, [[a8, b8], [c8]]], [b8, c8, [[a8, b8], [c8]]]], Yes, all_distinct_pairs_exhaustively_separated).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r03-hausdorff-space-0034/)

Hausdorff space · qf-mod42-hausdorff-full-audit · candidate





## 35. All Distinct Point Pairs Must Be Separated in the Complete Finite Topology

Use finite unique labeled points=[a9, b9, c9] and the complete canonical topology=[[a9, b9, c9], [a9, b9], [a9, c9], [a9], [b9, c9], [b9], [c9], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Give (points, pair_witness_rows, Hausdorff, certificate).

View answer and explanation([a9, b9, c9], [[a9, b9, [[a9, c9], [b9]]], [a9, c9, [[a9, b9], [c9]]], [b9, c9, [[a9, b9], [c9]]]], Yes, all_distinct_pairs_exhaustively_separated)

Enumerate every unordered pair in original point order and its first source-order witness. A single failed pair disproves Hausdorffness. The empty/singleton cases have no distinct pair and are vacuously Hausdorff. Finite Hausdorff implies discrete, but that corollary must not be extrapolated to infinite spaces. The exact result, retaining the requested order, is ([a9, b9, c9], [[a9, b9, [[a9, c9], [b9]]], [a9, c9, [[a9, b9], [c9]]], [b9, c9, [[a9, b9], [c9]]]], Yes, all_distinct_pairs_exhaustively_separated).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r03-hausdorff-space-0035/)

Hausdorff space · qf-mod42-hausdorff-full-audit · candidate





## 36. All Distinct Point Pairs Must Be Separated in the Complete Finite Topology

Use finite unique labeled points=[a10, b10, c10] and the complete canonical topology=[[a10, b10, c10], [a10, b10], [a10, c10], [a10], [b10, c10], [b10], [c10], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Give (points, pair_witness_rows, Hausdorff, certificate).

View answer and explanation([a10, b10, c10], [[a10, b10, [[a10, c10], [b10]]], [a10, c10, [[a10, b10], [c10]]], [b10, c10, [[a10, b10], [c10]]]], Yes, all_distinct_pairs_exhaustively_separated)

Enumerate every unordered pair in original point order and its first source-order witness. A single failed pair disproves Hausdorffness. The empty/singleton cases have no distinct pair and are vacuously Hausdorff. Finite Hausdorff implies discrete, but that corollary must not be extrapolated to infinite spaces. The exact result, retaining the requested order, is ([a10, b10, c10], [[a10, b10, [[a10, c10], [b10]]], [a10, c10, [[a10, b10], [c10]]], [b10, c10, [[a10, b10], [c10]]]], Yes, all_distinct_pairs_exhaustively_separated).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r03-hausdorff-space-0036/)

Hausdorff space · qf-mod42-hausdorff-full-audit · candidate





## 37. All Distinct Point Pairs Must Be Separated in the Complete Finite Topology

Use finite unique labeled points=[a11, b11, c11] and the complete canonical topology=[[a11, b11, c11], [a11, b11], [a11, c11], [a11], [b11, c11], [b11], [c11], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Give (points, pair_witness_rows, Hausdorff, certificate).

View answer and explanation([a11, b11, c11], [[a11, b11, [[a11, c11], [b11]]], [a11, c11, [[a11, b11], [c11]]], [b11, c11, [[a11, b11], [c11]]]], Yes, all_distinct_pairs_exhaustively_separated)

Enumerate every unordered pair in original point order and its first source-order witness. A single failed pair disproves Hausdorffness. The empty/singleton cases have no distinct pair and are vacuously Hausdorff. Finite Hausdorff implies discrete, but that corollary must not be extrapolated to infinite spaces. The exact result, retaining the requested order, is ([a11, b11, c11], [[a11, b11, [[a11, c11], [b11]]], [a11, c11, [[a11, b11], [c11]]], [b11, c11, [[a11, b11], [c11]]]], Yes, all_distinct_pairs_exhaustively_separated).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r03-hausdorff-space-0037/)

Hausdorff space · qf-mod42-hausdorff-full-audit · candidate





## 38. All Distinct Point Pairs Must Be Separated in the Complete Finite Topology

Use finite unique labeled points=[a12, b12, c12] and the complete canonical topology=[[a12, b12, c12], [a12, b12], [a12, c12], [a12], [b12, c12], [b12], [c12], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Give (points, pair_witness_rows, Hausdorff, certificate).

View answer and explanation([a12, b12, c12], [[a12, b12, [[a12, c12], [b12]]], [a12, c12, [[a12, b12], [c12]]], [b12, c12, [[a12, b12], [c12]]]], Yes, all_distinct_pairs_exhaustively_separated)

Enumerate every unordered pair in original point order and its first source-order witness. A single failed pair disproves Hausdorffness. The empty/singleton cases have no distinct pair and are vacuously Hausdorff. Finite Hausdorff implies discrete, but that corollary must not be extrapolated to infinite spaces. The exact result, retaining the requested order, is ([a12, b12, c12], [[a12, b12, [[a12, c12], [b12]]], [a12, c12, [[a12, b12], [c12]]], [b12, c12, [[a12, b12], [c12]]]], Yes, all_distinct_pairs_exhaustively_separated).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r03-hausdorff-space-0038/)

Hausdorff space · qf-mod42-hausdorff-full-audit · candidate





## 39. All Distinct Point Pairs Must Be Separated in the Complete Finite Topology

Use finite unique labeled points=[a13, b13, c13] and the complete canonical topology=[[a13, b13, c13], [a13, b13], [a13, c13], [a13], [b13, c13], [b13], [c13], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Give (points, pair_witness_rows, Hausdorff, certificate).

View answer and explanation([a13, b13, c13], [[a13, b13, [[a13, c13], [b13]]], [a13, c13, [[a13, b13], [c13]]], [b13, c13, [[a13, b13], [c13]]]], Yes, all_distinct_pairs_exhaustively_separated)

Enumerate every unordered pair in original point order and its first source-order witness. A single failed pair disproves Hausdorffness. The empty/singleton cases have no distinct pair and are vacuously Hausdorff. Finite Hausdorff implies discrete, but that corollary must not be extrapolated to infinite spaces. The exact result, retaining the requested order, is ([a13, b13, c13], [[a13, b13, [[a13, c13], [b13]]], [a13, c13, [[a13, b13], [c13]]], [b13, c13, [[a13, b13], [c13]]]], Yes, all_distinct_pairs_exhaustively_separated).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r03-hausdorff-space-0039/)

Hausdorff space · qf-mod42-hausdorff-full-audit · candidate





## 40. All Distinct Point Pairs Must Be Separated in the Complete Finite Topology

Use finite unique labeled points=[a14, b14, c14] and the complete canonical topology=[[a14, b14, c14], [a14, b14], [a14, c14], [a14], [b14, c14], [b14], [c14], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Give (points, pair_witness_rows, Hausdorff, certificate).

View answer and explanation([a14, b14, c14], [[a14, b14, [[a14, c14], [b14]]], [a14, c14, [[a14, b14], [c14]]], [b14, c14, [[a14, b14], [c14]]]], Yes, all_distinct_pairs_exhaustively_separated)

Enumerate every unordered pair in original point order and its first source-order witness. A single failed pair disproves Hausdorffness. The empty/singleton cases have no distinct pair and are vacuously Hausdorff. Finite Hausdorff implies discrete, but that corollary must not be extrapolated to infinite spaces. The exact result, retaining the requested order, is ([a14, b14, c14], [[a14, b14, [[a14, c14], [b14]]], [a14, c14, [[a14, b14], [c14]]], [b14, c14, [[a14, b14], [c14]]]], Yes, all_distinct_pairs_exhaustively_separated).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r03-hausdorff-space-0040/)

Hausdorff space · qf-mod42-hausdorff-full-audit · candidate





## 41. All Distinct Point Pairs Must Be Separated in the Complete Finite Topology

Use finite unique labeled points=[a15, b15, c15] and the complete canonical topology=[[a15, b15, c15], [a15, b15], [a15, c15], [a15], [b15, c15], [b15], [c15], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Give (points, pair_witness_rows, Hausdorff, certificate).

View answer and explanation([a15, b15, c15], [[a15, b15, [[a15, c15], [b15]]], [a15, c15, [[a15, b15], [c15]]], [b15, c15, [[a15, b15], [c15]]]], Yes, all_distinct_pairs_exhaustively_separated)

Enumerate every unordered pair in original point order and its first source-order witness. A single failed pair disproves Hausdorffness. The empty/singleton cases have no distinct pair and are vacuously Hausdorff. Finite Hausdorff implies discrete, but that corollary must not be extrapolated to infinite spaces. The exact result, retaining the requested order, is ([a15, b15, c15], [[a15, b15, [[a15, c15], [b15]]], [a15, c15, [[a15, b15], [c15]]], [b15, c15, [[a15, b15], [c15]]]], Yes, all_distinct_pairs_exhaustively_separated).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r03-hausdorff-space-0041/)

Hausdorff space · qf-mod42-hausdorff-full-audit · candidate





## 42. All Distinct Point Pairs Must Be Separated in the Complete Finite Topology

Use finite unique labeled points=[a16, b16, c16] and the complete canonical topology=[[a16, b16, c16], [a16, b16], [a16, c16], [a16], [b16, c16], [b16], [c16], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Give (points, pair_witness_rows, Hausdorff, certificate).

View answer and explanation([a16, b16, c16], [[a16, b16, [[a16, c16], [b16]]], [a16, c16, [[a16, b16], [c16]]], [b16, c16, [[a16, b16], [c16]]]], Yes, all_distinct_pairs_exhaustively_separated)

Enumerate every unordered pair in original point order and its first source-order witness. A single failed pair disproves Hausdorffness. The empty/singleton cases have no distinct pair and are vacuously Hausdorff. Finite Hausdorff implies discrete, but that corollary must not be extrapolated to infinite spaces. The exact result, retaining the requested order, is ([a16, b16, c16], [[a16, b16, [[a16, c16], [b16]]], [a16, c16, [[a16, b16], [c16]]], [b16, c16, [[a16, b16], [c16]]]], Yes, all_distinct_pairs_exhaustively_separated).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r03-hausdorff-space-0042/)

Hausdorff space · qf-mod42-hausdorff-full-audit · candidate





## 43. All Distinct Point Pairs Must Be Separated in the Complete Finite Topology

Use finite unique labeled points=[a17, b17, c17] and the complete canonical topology=[[a17, b17, c17], [a17, b17], [a17, c17], [a17], [b17, c17], [b17], [c17], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Give (points, pair_witness_rows, Hausdorff, certificate).

View answer and explanation([a17, b17, c17], [[a17, b17, [[a17, c17], [b17]]], [a17, c17, [[a17, b17], [c17]]], [b17, c17, [[a17, b17], [c17]]]], Yes, all_distinct_pairs_exhaustively_separated)

Enumerate every unordered pair in original point order and its first source-order witness. A single failed pair disproves Hausdorffness. The empty/singleton cases have no distinct pair and are vacuously Hausdorff. Finite Hausdorff implies discrete, but that corollary must not be extrapolated to infinite spaces. The exact result, retaining the requested order, is ([a17, b17, c17], [[a17, b17, [[a17, c17], [b17]]], [a17, c17, [[a17, b17], [c17]]], [b17, c17, [[a17, b17], [c17]]]], Yes, all_distinct_pairs_exhaustively_separated).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r03-hausdorff-space-0043/)

Hausdorff space · qf-mod42-hausdorff-full-audit · candidate





## 44. All Distinct Point Pairs Must Be Separated in the Complete Finite Topology

Use finite unique labeled points=[a18, b18, c18] and the complete canonical topology=[[a18, b18, c18], [a18, b18], [a18, c18], [a18], [b18, c18], [b18], [c18], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Give (points, pair_witness_rows, Hausdorff, certificate).

View answer and explanation([a18, b18, c18], [[a18, b18, [[a18, c18], [b18]]], [a18, c18, [[a18, b18], [c18]]], [b18, c18, [[a18, b18], [c18]]]], Yes, all_distinct_pairs_exhaustively_separated)

Enumerate every unordered pair in original point order and its first source-order witness. A single failed pair disproves Hausdorffness. The empty/singleton cases have no distinct pair and are vacuously Hausdorff. Finite Hausdorff implies discrete, but that corollary must not be extrapolated to infinite spaces. The exact result, retaining the requested order, is ([a18, b18, c18], [[a18, b18, [[a18, c18], [b18]]], [a18, c18, [[a18, b18], [c18]]], [b18, c18, [[a18, b18], [c18]]]], Yes, all_distinct_pairs_exhaustively_separated).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r03-hausdorff-space-0044/)

Hausdorff space · qf-mod42-hausdorff-full-audit · candidate





## 45. All Distinct Point Pairs Must Be Separated in the Complete Finite Topology

Use finite unique labeled points=[a19, b19, c19] and the complete canonical topology=[[a19, b19, c19], [a19, b19], [a19, c19], [a19], [b19, c19], [b19], [c19], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Give (points, pair_witness_rows, Hausdorff, certificate).

View answer and explanation([a19, b19, c19], [[a19, b19, [[a19, c19], [b19]]], [a19, c19, [[a19, b19], [c19]]], [b19, c19, [[a19, b19], [c19]]]], Yes, all_distinct_pairs_exhaustively_separated)

Enumerate every unordered pair in original point order and its first source-order witness. A single failed pair disproves Hausdorffness. The empty/singleton cases have no distinct pair and are vacuously Hausdorff. Finite Hausdorff implies discrete, but that corollary must not be extrapolated to infinite spaces. The exact result, retaining the requested order, is ([a19, b19, c19], [[a19, b19, [[a19, c19], [b19]]], [a19, c19, [[a19, b19], [c19]]], [b19, c19, [[a19, b19], [c19]]]], Yes, all_distinct_pairs_exhaustively_separated).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r03-hausdorff-space-0045/)

Hausdorff space · qf-mod42-hausdorff-full-audit · candidate





## 46. All Distinct Point Pairs Must Be Separated in the Complete Finite Topology

Use finite unique labeled points=[a20, b20, c20] and the complete canonical topology=[[a20, b20, c20], [a20, b20], [a20, c20], [a20], [b20, c20], [b20], [c20], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Give (points, pair_witness_rows, Hausdorff, certificate).

View answer and explanation([a20, b20, c20], [[a20, b20, [[a20, c20], [b20]]], [a20, c20, [[a20, b20], [c20]]], [b20, c20, [[a20, b20], [c20]]]], Yes, all_distinct_pairs_exhaustively_separated)

Enumerate every unordered pair in original point order and its first source-order witness. A single failed pair disproves Hausdorffness. The empty/singleton cases have no distinct pair and are vacuously Hausdorff. Finite Hausdorff implies discrete, but that corollary must not be extrapolated to infinite spaces. The exact result, retaining the requested order, is ([a20, b20, c20], [[a20, b20, [[a20, c20], [b20]]], [a20, c20, [[a20, b20], [c20]]], [b20, c20, [[a20, b20], [c20]]]], Yes, all_distinct_pairs_exhaustively_separated).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r03-hausdorff-space-0046/)

Hausdorff space · qf-mod42-hausdorff-full-audit · candidate





## 47. All Distinct Point Pairs Must Be Separated in the Complete Finite Topology

Use finite unique labeled points=[a21, b21, c21] and the complete canonical topology=[[a21, b21, c21], [a21, b21], [a21, c21], [a21], [b21, c21], [b21], [c21], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Give (points, pair_witness_rows, Hausdorff, certificate).

View answer and explanation([a21, b21, c21], [[a21, b21, [[a21, c21], [b21]]], [a21, c21, [[a21, b21], [c21]]], [b21, c21, [[a21, b21], [c21]]]], Yes, all_distinct_pairs_exhaustively_separated)

Enumerate every unordered pair in original point order and its first source-order witness. A single failed pair disproves Hausdorffness. The empty/singleton cases have no distinct pair and are vacuously Hausdorff. Finite Hausdorff implies discrete, but that corollary must not be extrapolated to infinite spaces. The exact result, retaining the requested order, is ([a21, b21, c21], [[a21, b21, [[a21, c21], [b21]]], [a21, c21, [[a21, b21], [c21]]], [b21, c21, [[a21, b21], [c21]]]], Yes, all_distinct_pairs_exhaustively_separated).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r03-hausdorff-space-0047/)

Hausdorff space · qf-mod42-hausdorff-full-audit · candidate





## 48. All Distinct Point Pairs Must Be Separated in the Complete Finite Topology

Use finite unique labeled points=[a22, b22, c22] and the complete canonical topology=[[a22, b22, c22], [a22, b22], [a22, c22], [a22], [b22, c22], [b22], [c22], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Give (points, pair_witness_rows, Hausdorff, certificate).

View answer and explanation([a22, b22, c22], [[a22, b22, [[a22, c22], [b22]]], [a22, c22, [[a22, b22], [c22]]], [b22, c22, [[a22, b22], [c22]]]], Yes, all_distinct_pairs_exhaustively_separated)

Enumerate every unordered pair in original point order and its first source-order witness. A single failed pair disproves Hausdorffness. The empty/singleton cases have no distinct pair and are vacuously Hausdorff. Finite Hausdorff implies discrete, but that corollary must not be extrapolated to infinite spaces. The exact result, retaining the requested order, is ([a22, b22, c22], [[a22, b22, [[a22, c22], [b22]]], [a22, c22, [[a22, b22], [c22]]], [b22, c22, [[a22, b22], [c22]]]], Yes, all_distinct_pairs_exhaustively_separated).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r03-hausdorff-space-0048/)

Hausdorff space · qf-mod42-hausdorff-full-audit · candidate





## 49. All Distinct Point Pairs Must Be Separated in the Complete Finite Topology

Use finite unique labeled points=[a23, b23, c23] and the complete canonical topology=[[a23, b23, c23], [a23, b23], [a23, c23], [a23], [b23, c23], [b23], [c23], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Give (points, pair_witness_rows, Hausdorff, certificate).

View answer and explanation([a23, b23, c23], [[a23, b23, [[a23, c23], [b23]]], [a23, c23, [[a23, b23], [c23]]], [b23, c23, [[a23, b23], [c23]]]], Yes, all_distinct_pairs_exhaustively_separated)

Enumerate every unordered pair in original point order and its first source-order witness. A single failed pair disproves Hausdorffness. The empty/singleton cases have no distinct pair and are vacuously Hausdorff. Finite Hausdorff implies discrete, but that corollary must not be extrapolated to infinite spaces. The exact result, retaining the requested order, is ([a23, b23, c23], [[a23, b23, [[a23, c23], [b23]]], [a23, c23, [[a23, b23], [c23]]], [b23, c23, [[a23, b23], [c23]]]], Yes, all_distinct_pairs_exhaustively_separated).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r03-hausdorff-space-0049/)

Hausdorff space · qf-mod42-hausdorff-full-audit · candidate





## 50. All Distinct Point Pairs Must Be Separated in the Complete Finite Topology

Use finite unique labeled points=[a24, b24, c24] and the complete canonical topology=[[a24, b24, c24], [a24, b24], [a24, c24], [a24], [b24, c24], [b24], [c24], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Give (points, pair_witness_rows, Hausdorff, certificate).

View answer and explanation([a24, b24, c24], [[a24, b24, [[a24, c24], [b24]]], [a24, c24, [[a24, b24], [c24]]], [b24, c24, [[a24, b24], [c24]]]], Yes, all_distinct_pairs_exhaustively_separated)

Enumerate every unordered pair in original point order and its first source-order witness. A single failed pair disproves Hausdorffness. The empty/singleton cases have no distinct pair and are vacuously Hausdorff. Finite Hausdorff implies discrete, but that corollary must not be extrapolated to infinite spaces. The exact result, retaining the requested order, is ([a24, b24, c24], [[a24, b24, [[a24, c24], [b24]]], [a24, c24, [[a24, b24], [c24]]], [b24, c24, [[a24, b24], [c24]]]], Yes, all_distinct_pairs_exhaustively_separated).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r03-hausdorff-space-0050/)

Hausdorff space · qf-mod42-hausdorff-full-audit · candidate





## 51. One Exhaustively Failed Distinct Pair Proves Non-Hausdorffness

Use finite unique labeled points=[a0, b0, c0] and the complete canonical topology=[[a0, b0, c0], [a0, b0], [a0], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Test x=a0, y=b0. Give (x, y, witness_or_null, separated, boundary).

View answer and explanation(a0, b0, null, No, one_failed_distinct_pair_proves_non_Hausdorff)

Require the full open-pair search to find no separator. Report null and No separation; this complete failed-pair certificate differs from simply not seeing a witness in a sample. The exact result, retaining the requested order, is (a0, b0, null, No, one_failed_distinct_pair_proves_non_Hausdorff).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r03-hausdorff-space-0051/)

Hausdorff space · qf-mod42-hausdorff-inseparable-pair · candidate





## 52. One Exhaustively Failed Distinct Pair Proves Non-Hausdorffness

Use finite unique labeled points=[a1, b1, c1] and the complete canonical topology=[[a1, b1, c1], [a1, b1], [a1], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Test x=a1, y=b1. Give (x, y, witness_or_null, separated, boundary).

View answer and explanation(a1, b1, null, No, one_failed_distinct_pair_proves_non_Hausdorff)

Require the full open-pair search to find no separator. Report null and No separation; this complete failed-pair certificate differs from simply not seeing a witness in a sample. The exact result, retaining the requested order, is (a1, b1, null, No, one_failed_distinct_pair_proves_non_Hausdorff).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r03-hausdorff-space-0052/)

Hausdorff space · qf-mod42-hausdorff-inseparable-pair · candidate





## 53. One Exhaustively Failed Distinct Pair Proves Non-Hausdorffness

Use finite unique labeled points=[a2, b2, c2] and the complete canonical topology=[[a2, b2, c2], [a2, b2], [a2], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Test x=a2, y=b2. Give (x, y, witness_or_null, separated, boundary).

View answer and explanation(a2, b2, null, No, one_failed_distinct_pair_proves_non_Hausdorff)

Require the full open-pair search to find no separator. Report null and No separation; this complete failed-pair certificate differs from simply not seeing a witness in a sample. The exact result, retaining the requested order, is (a2, b2, null, No, one_failed_distinct_pair_proves_non_Hausdorff).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r03-hausdorff-space-0053/)

Hausdorff space · qf-mod42-hausdorff-inseparable-pair · candidate





## 54. One Exhaustively Failed Distinct Pair Proves Non-Hausdorffness

Use finite unique labeled points=[a3, b3, c3] and the complete canonical topology=[[a3, b3, c3], [a3, b3], [a3], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Test x=a3, y=b3. Give (x, y, witness_or_null, separated, boundary).

View answer and explanation(a3, b3, null, No, one_failed_distinct_pair_proves_non_Hausdorff)

Require the full open-pair search to find no separator. Report null and No separation; this complete failed-pair certificate differs from simply not seeing a witness in a sample. The exact result, retaining the requested order, is (a3, b3, null, No, one_failed_distinct_pair_proves_non_Hausdorff).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r03-hausdorff-space-0054/)

Hausdorff space · qf-mod42-hausdorff-inseparable-pair · candidate





## 55. One Exhaustively Failed Distinct Pair Proves Non-Hausdorffness

Use finite unique labeled points=[a4, b4, c4] and the complete canonical topology=[[a4, b4, c4], [a4, b4], [a4], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Test x=a4, y=b4. Give (x, y, witness_or_null, separated, boundary).

View answer and explanation(a4, b4, null, No, one_failed_distinct_pair_proves_non_Hausdorff)

Require the full open-pair search to find no separator. Report null and No separation; this complete failed-pair certificate differs from simply not seeing a witness in a sample. The exact result, retaining the requested order, is (a4, b4, null, No, one_failed_distinct_pair_proves_non_Hausdorff).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r03-hausdorff-space-0055/)

Hausdorff space · qf-mod42-hausdorff-inseparable-pair · candidate





## 56. One Exhaustively Failed Distinct Pair Proves Non-Hausdorffness

Use finite unique labeled points=[a5, b5, c5] and the complete canonical topology=[[a5, b5, c5], [a5, b5], [a5], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Test x=a5, y=b5. Give (x, y, witness_or_null, separated, boundary).

View answer and explanation(a5, b5, null, No, one_failed_distinct_pair_proves_non_Hausdorff)

Require the full open-pair search to find no separator. Report null and No separation; this complete failed-pair certificate differs from simply not seeing a witness in a sample. The exact result, retaining the requested order, is (a5, b5, null, No, one_failed_distinct_pair_proves_non_Hausdorff).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r03-hausdorff-space-0056/)

Hausdorff space · qf-mod42-hausdorff-inseparable-pair · candidate





## 57. One Exhaustively Failed Distinct Pair Proves Non-Hausdorffness

Use finite unique labeled points=[a6, b6, c6] and the complete canonical topology=[[a6, b6, c6], [a6, b6], [a6], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Test x=a6, y=b6. Give (x, y, witness_or_null, separated, boundary).

View answer and explanation(a6, b6, null, No, one_failed_distinct_pair_proves_non_Hausdorff)

Require the full open-pair search to find no separator. Report null and No separation; this complete failed-pair certificate differs from simply not seeing a witness in a sample. The exact result, retaining the requested order, is (a6, b6, null, No, one_failed_distinct_pair_proves_non_Hausdorff).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r03-hausdorff-space-0057/)

Hausdorff space · qf-mod42-hausdorff-inseparable-pair · candidate





## 58. One Exhaustively Failed Distinct Pair Proves Non-Hausdorffness

Use finite unique labeled points=[a7, b7, c7] and the complete canonical topology=[[a7, b7, c7], [a7, b7], [a7], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Test x=a7, y=b7. Give (x, y, witness_or_null, separated, boundary).

View answer and explanation(a7, b7, null, No, one_failed_distinct_pair_proves_non_Hausdorff)

Require the full open-pair search to find no separator. Report null and No separation; this complete failed-pair certificate differs from simply not seeing a witness in a sample. The exact result, retaining the requested order, is (a7, b7, null, No, one_failed_distinct_pair_proves_non_Hausdorff).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r03-hausdorff-space-0058/)

Hausdorff space · qf-mod42-hausdorff-inseparable-pair · candidate





## 59. One Exhaustively Failed Distinct Pair Proves Non-Hausdorffness

Use finite unique labeled points=[a8, b8, c8] and the complete canonical topology=[[a8, b8, c8], [a8, b8], [a8], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Test x=a8, y=b8. Give (x, y, witness_or_null, separated, boundary).

View answer and explanation(a8, b8, null, No, one_failed_distinct_pair_proves_non_Hausdorff)

Require the full open-pair search to find no separator. Report null and No separation; this complete failed-pair certificate differs from simply not seeing a witness in a sample. The exact result, retaining the requested order, is (a8, b8, null, No, one_failed_distinct_pair_proves_non_Hausdorff).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r03-hausdorff-space-0059/)

Hausdorff space · qf-mod42-hausdorff-inseparable-pair · candidate





## 60. One Exhaustively Failed Distinct Pair Proves Non-Hausdorffness

Use finite unique labeled points=[a9, b9, c9] and the complete canonical topology=[[a9, b9, c9], [a9, b9], [a9], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Test x=a9, y=b9. Give (x, y, witness_or_null, separated, boundary).

View answer and explanation(a9, b9, null, No, one_failed_distinct_pair_proves_non_Hausdorff)

Require the full open-pair search to find no separator. Report null and No separation; this complete failed-pair certificate differs from simply not seeing a witness in a sample. The exact result, retaining the requested order, is (a9, b9, null, No, one_failed_distinct_pair_proves_non_Hausdorff).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r03-hausdorff-space-0060/)

Hausdorff space · qf-mod42-hausdorff-inseparable-pair · candidate





## 61. One Exhaustively Failed Distinct Pair Proves Non-Hausdorffness

Use finite unique labeled points=[a10, b10, c10] and the complete canonical topology=[[a10, b10, c10], [a10, b10], [a10], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Test x=a10, y=b10. Give (x, y, witness_or_null, separated, boundary).

View answer and explanation(a10, b10, null, No, one_failed_distinct_pair_proves_non_Hausdorff)

Require the full open-pair search to find no separator. Report null and No separation; this complete failed-pair certificate differs from simply not seeing a witness in a sample. The exact result, retaining the requested order, is (a10, b10, null, No, one_failed_distinct_pair_proves_non_Hausdorff).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r03-hausdorff-space-0061/)

Hausdorff space · qf-mod42-hausdorff-inseparable-pair · candidate





## 62. One Exhaustively Failed Distinct Pair Proves Non-Hausdorffness

Use finite unique labeled points=[a11, b11, c11] and the complete canonical topology=[[a11, b11, c11], [a11, b11], [a11], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Test x=a11, y=b11. Give (x, y, witness_or_null, separated, boundary).

View answer and explanation(a11, b11, null, No, one_failed_distinct_pair_proves_non_Hausdorff)

Require the full open-pair search to find no separator. Report null and No separation; this complete failed-pair certificate differs from simply not seeing a witness in a sample. The exact result, retaining the requested order, is (a11, b11, null, No, one_failed_distinct_pair_proves_non_Hausdorff).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r03-hausdorff-space-0062/)

Hausdorff space · qf-mod42-hausdorff-inseparable-pair · candidate





## 63. One Exhaustively Failed Distinct Pair Proves Non-Hausdorffness

Use finite unique labeled points=[a12, b12, c12] and the complete canonical topology=[[a12, b12, c12], [a12, b12], [a12], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Test x=a12, y=b12. Give (x, y, witness_or_null, separated, boundary).

View answer and explanation(a12, b12, null, No, one_failed_distinct_pair_proves_non_Hausdorff)

Require the full open-pair search to find no separator. Report null and No separation; this complete failed-pair certificate differs from simply not seeing a witness in a sample. The exact result, retaining the requested order, is (a12, b12, null, No, one_failed_distinct_pair_proves_non_Hausdorff).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r03-hausdorff-space-0063/)

Hausdorff space · qf-mod42-hausdorff-inseparable-pair · candidate





## 64. One Exhaustively Failed Distinct Pair Proves Non-Hausdorffness

Use finite unique labeled points=[a13, b13, c13] and the complete canonical topology=[[a13, b13, c13], [a13, b13], [a13], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Test x=a13, y=b13. Give (x, y, witness_or_null, separated, boundary).

View answer and explanation(a13, b13, null, No, one_failed_distinct_pair_proves_non_Hausdorff)

Require the full open-pair search to find no separator. Report null and No separation; this complete failed-pair certificate differs from simply not seeing a witness in a sample. The exact result, retaining the requested order, is (a13, b13, null, No, one_failed_distinct_pair_proves_non_Hausdorff).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r03-hausdorff-space-0064/)

Hausdorff space · qf-mod42-hausdorff-inseparable-pair · candidate





## 65. One Exhaustively Failed Distinct Pair Proves Non-Hausdorffness

Use finite unique labeled points=[a14, b14, c14] and the complete canonical topology=[[a14, b14, c14], [a14, b14], [a14], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Test x=a14, y=b14. Give (x, y, witness_or_null, separated, boundary).

View answer and explanation(a14, b14, null, No, one_failed_distinct_pair_proves_non_Hausdorff)

Require the full open-pair search to find no separator. Report null and No separation; this complete failed-pair certificate differs from simply not seeing a witness in a sample. The exact result, retaining the requested order, is (a14, b14, null, No, one_failed_distinct_pair_proves_non_Hausdorff).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r03-hausdorff-space-0065/)

Hausdorff space · qf-mod42-hausdorff-inseparable-pair · candidate





## 66. One Exhaustively Failed Distinct Pair Proves Non-Hausdorffness

Use finite unique labeled points=[a15, b15, c15] and the complete canonical topology=[[a15, b15, c15], [a15, b15], [a15], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Test x=a15, y=b15. Give (x, y, witness_or_null, separated, boundary).

View answer and explanation(a15, b15, null, No, one_failed_distinct_pair_proves_non_Hausdorff)

Require the full open-pair search to find no separator. Report null and No separation; this complete failed-pair certificate differs from simply not seeing a witness in a sample. The exact result, retaining the requested order, is (a15, b15, null, No, one_failed_distinct_pair_proves_non_Hausdorff).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r03-hausdorff-space-0066/)

Hausdorff space · qf-mod42-hausdorff-inseparable-pair · candidate





## 67. One Exhaustively Failed Distinct Pair Proves Non-Hausdorffness

Use finite unique labeled points=[a16, b16, c16] and the complete canonical topology=[[a16, b16, c16], [a16, b16], [a16], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Test x=a16, y=b16. Give (x, y, witness_or_null, separated, boundary).

View answer and explanation(a16, b16, null, No, one_failed_distinct_pair_proves_non_Hausdorff)

Require the full open-pair search to find no separator. Report null and No separation; this complete failed-pair certificate differs from simply not seeing a witness in a sample. The exact result, retaining the requested order, is (a16, b16, null, No, one_failed_distinct_pair_proves_non_Hausdorff).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r03-hausdorff-space-0067/)

Hausdorff space · qf-mod42-hausdorff-inseparable-pair · candidate





## 68. One Exhaustively Failed Distinct Pair Proves Non-Hausdorffness

Use finite unique labeled points=[a17, b17, c17] and the complete canonical topology=[[a17, b17, c17], [a17, b17], [a17], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Test x=a17, y=b17. Give (x, y, witness_or_null, separated, boundary).

View answer and explanation(a17, b17, null, No, one_failed_distinct_pair_proves_non_Hausdorff)

Require the full open-pair search to find no separator. Report null and No separation; this complete failed-pair certificate differs from simply not seeing a witness in a sample. The exact result, retaining the requested order, is (a17, b17, null, No, one_failed_distinct_pair_proves_non_Hausdorff).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r03-hausdorff-space-0068/)

Hausdorff space · qf-mod42-hausdorff-inseparable-pair · candidate





## 69. One Exhaustively Failed Distinct Pair Proves Non-Hausdorffness

Use finite unique labeled points=[a18, b18, c18] and the complete canonical topology=[[a18, b18, c18], [a18, b18], [a18], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Test x=a18, y=b18. Give (x, y, witness_or_null, separated, boundary).

View answer and explanation(a18, b18, null, No, one_failed_distinct_pair_proves_non_Hausdorff)

Require the full open-pair search to find no separator. Report null and No separation; this complete failed-pair certificate differs from simply not seeing a witness in a sample. The exact result, retaining the requested order, is (a18, b18, null, No, one_failed_distinct_pair_proves_non_Hausdorff).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r03-hausdorff-space-0069/)

Hausdorff space · qf-mod42-hausdorff-inseparable-pair · candidate





## 70. One Exhaustively Failed Distinct Pair Proves Non-Hausdorffness

Use finite unique labeled points=[a19, b19, c19] and the complete canonical topology=[[a19, b19, c19], [a19, b19], [a19], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Test x=a19, y=b19. Give (x, y, witness_or_null, separated, boundary).

View answer and explanation(a19, b19, null, No, one_failed_distinct_pair_proves_non_Hausdorff)

Require the full open-pair search to find no separator. Report null and No separation; this complete failed-pair certificate differs from simply not seeing a witness in a sample. The exact result, retaining the requested order, is (a19, b19, null, No, one_failed_distinct_pair_proves_non_Hausdorff).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r03-hausdorff-space-0070/)

Hausdorff space · qf-mod42-hausdorff-inseparable-pair · candidate





## 71. One Exhaustively Failed Distinct Pair Proves Non-Hausdorffness

Use finite unique labeled points=[a20, b20, c20] and the complete canonical topology=[[a20, b20, c20], [a20, b20], [a20], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Test x=a20, y=b20. Give (x, y, witness_or_null, separated, boundary).

View answer and explanation(a20, b20, null, No, one_failed_distinct_pair_proves_non_Hausdorff)

Require the full open-pair search to find no separator. Report null and No separation; this complete failed-pair certificate differs from simply not seeing a witness in a sample. The exact result, retaining the requested order, is (a20, b20, null, No, one_failed_distinct_pair_proves_non_Hausdorff).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r03-hausdorff-space-0071/)

Hausdorff space · qf-mod42-hausdorff-inseparable-pair · candidate





## 72. One Exhaustively Failed Distinct Pair Proves Non-Hausdorffness

Use finite unique labeled points=[a21, b21, c21] and the complete canonical topology=[[a21, b21, c21], [a21, b21], [a21], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Test x=a21, y=b21. Give (x, y, witness_or_null, separated, boundary).

View answer and explanation(a21, b21, null, No, one_failed_distinct_pair_proves_non_Hausdorff)

Require the full open-pair search to find no separator. Report null and No separation; this complete failed-pair certificate differs from simply not seeing a witness in a sample. The exact result, retaining the requested order, is (a21, b21, null, No, one_failed_distinct_pair_proves_non_Hausdorff).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r03-hausdorff-space-0072/)

Hausdorff space · qf-mod42-hausdorff-inseparable-pair · candidate





## 73. One Exhaustively Failed Distinct Pair Proves Non-Hausdorffness

Use finite unique labeled points=[a22, b22, c22] and the complete canonical topology=[[a22, b22, c22], [a22, b22], [a22], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Test x=a22, y=b22. Give (x, y, witness_or_null, separated, boundary).

View answer and explanation(a22, b22, null, No, one_failed_distinct_pair_proves_non_Hausdorff)

Require the full open-pair search to find no separator. Report null and No separation; this complete failed-pair certificate differs from simply not seeing a witness in a sample. The exact result, retaining the requested order, is (a22, b22, null, No, one_failed_distinct_pair_proves_non_Hausdorff).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r03-hausdorff-space-0073/)

Hausdorff space · qf-mod42-hausdorff-inseparable-pair · candidate





## 74. One Exhaustively Failed Distinct Pair Proves Non-Hausdorffness

Use finite unique labeled points=[a23, b23, c23] and the complete canonical topology=[[a23, b23, c23], [a23, b23], [a23], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Test x=a23, y=b23. Give (x, y, witness_or_null, separated, boundary).

View answer and explanation(a23, b23, null, No, one_failed_distinct_pair_proves_non_Hausdorff)

Require the full open-pair search to find no separator. Report null and No separation; this complete failed-pair certificate differs from simply not seeing a witness in a sample. The exact result, retaining the requested order, is (a23, b23, null, No, one_failed_distinct_pair_proves_non_Hausdorff).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r03-hausdorff-space-0074/)

Hausdorff space · qf-mod42-hausdorff-inseparable-pair · candidate





## 75. One Exhaustively Failed Distinct Pair Proves Non-Hausdorffness

Use finite unique labeled points=[a24, b24, c24] and the complete canonical topology=[[a24, b24, c24], [a24, b24], [a24], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Test x=a24, y=b24. Give (x, y, witness_or_null, separated, boundary).

View answer and explanation(a24, b24, null, No, one_failed_distinct_pair_proves_non_Hausdorff)

Require the full open-pair search to find no separator. Report null and No separation; this complete failed-pair certificate differs from simply not seeing a witness in a sample. The exact result, retaining the requested order, is (a24, b24, null, No, one_failed_distinct_pair_proves_non_Hausdorff).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r03-hausdorff-space-0075/)

Hausdorff space · qf-mod42-hausdorff-inseparable-pair · candidate





## 76. Successful Pair Samples Do Not Certify Unchecked or Infinite Cases

Use finite unique labeled points=[a0, b0, c0] and the complete canonical topology=[[a0, b0, c0], [a0, b0], [a0, c0], [a0], [c0], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Supplied sampled_pairs=[[c0, a0]]. Give (sampled_pair_rows, all_sampled_separated, sample_count, total_pair_count, general_sample_certificate, boundary).

The fixed No withholds a general sampled-space or unchecked-pair certificate; it does not negate the separation flags. If all finite pairs were actually covered, that completeness must be stated separately.

View answer and explanation([[c0, a0, [[c0], [a0, b0]]]], Yes, 1, 3, No, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)

Audit each sampled pair and retain sample count and total finite unordered-pair count separately. The fixed No withholds a general sampling certificate, not the preceding separation results. If every finite pair was actually covered, completeness must be established separately rather than inferred from the word sample. The exact result, retaining the requested order, is ([[c0, a0, [[c0], [a0, b0]]]], Yes, 1, 3, No, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r03-hausdorff-space-0076/)

Hausdorff space · qf-mod42-hausdorff-sampled-boundary · candidate





## 77. Successful Pair Samples Do Not Certify Unchecked or Infinite Cases

Use finite unique labeled points=[a1, b1, c1] and the complete canonical topology=[[a1, b1, c1], [a1, b1], [a1, c1], [a1], [c1], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Supplied sampled_pairs=[[c1, a1]]. Give (sampled_pair_rows, all_sampled_separated, sample_count, total_pair_count, general_sample_certificate, boundary).

The fixed No withholds a general sampled-space or unchecked-pair certificate; it does not negate the separation flags. If all finite pairs were actually covered, that completeness must be stated separately.

View answer and explanation([[c1, a1, [[c1], [a1, b1]]]], Yes, 1, 3, No, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)

Audit each sampled pair and retain sample count and total finite unordered-pair count separately. The fixed No withholds a general sampling certificate, not the preceding separation results. If every finite pair was actually covered, completeness must be established separately rather than inferred from the word sample. The exact result, retaining the requested order, is ([[c1, a1, [[c1], [a1, b1]]]], Yes, 1, 3, No, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r03-hausdorff-space-0077/)

Hausdorff space · qf-mod42-hausdorff-sampled-boundary · candidate





## 78. Successful Pair Samples Do Not Certify Unchecked or Infinite Cases

Use finite unique labeled points=[a2, b2, c2] and the complete canonical topology=[[a2, b2, c2], [a2, b2], [a2, c2], [a2], [c2], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Supplied sampled_pairs=[[c2, a2]]. Give (sampled_pair_rows, all_sampled_separated, sample_count, total_pair_count, general_sample_certificate, boundary).

The fixed No withholds a general sampled-space or unchecked-pair certificate; it does not negate the separation flags. If all finite pairs were actually covered, that completeness must be stated separately.

View answer and explanation([[c2, a2, [[c2], [a2, b2]]]], Yes, 1, 3, No, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)

Audit each sampled pair and retain sample count and total finite unordered-pair count separately. The fixed No withholds a general sampling certificate, not the preceding separation results. If every finite pair was actually covered, completeness must be established separately rather than inferred from the word sample. The exact result, retaining the requested order, is ([[c2, a2, [[c2], [a2, b2]]]], Yes, 1, 3, No, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r03-hausdorff-space-0078/)

Hausdorff space · qf-mod42-hausdorff-sampled-boundary · candidate





## 79. Successful Pair Samples Do Not Certify Unchecked or Infinite Cases

Use finite unique labeled points=[a3, b3, c3] and the complete canonical topology=[[a3, b3, c3], [a3, b3], [a3, c3], [a3], [c3], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Supplied sampled_pairs=[[c3, a3]]. Give (sampled_pair_rows, all_sampled_separated, sample_count, total_pair_count, general_sample_certificate, boundary).

The fixed No withholds a general sampled-space or unchecked-pair certificate; it does not negate the separation flags. If all finite pairs were actually covered, that completeness must be stated separately.

View answer and explanation([[c3, a3, [[c3], [a3, b3]]]], Yes, 1, 3, No, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)

Audit each sampled pair and retain sample count and total finite unordered-pair count separately. The fixed No withholds a general sampling certificate, not the preceding separation results. If every finite pair was actually covered, completeness must be established separately rather than inferred from the word sample. The exact result, retaining the requested order, is ([[c3, a3, [[c3], [a3, b3]]]], Yes, 1, 3, No, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r03-hausdorff-space-0079/)

Hausdorff space · qf-mod42-hausdorff-sampled-boundary · candidate





## 80. Successful Pair Samples Do Not Certify Unchecked or Infinite Cases

Use finite unique labeled points=[a4, b4, c4] and the complete canonical topology=[[a4, b4, c4], [a4, b4], [a4, c4], [a4], [c4], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Supplied sampled_pairs=[[c4, a4]]. Give (sampled_pair_rows, all_sampled_separated, sample_count, total_pair_count, general_sample_certificate, boundary).

The fixed No withholds a general sampled-space or unchecked-pair certificate; it does not negate the separation flags. If all finite pairs were actually covered, that completeness must be stated separately.

View answer and explanation([[c4, a4, [[c4], [a4, b4]]]], Yes, 1, 3, No, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)

Audit each sampled pair and retain sample count and total finite unordered-pair count separately. The fixed No withholds a general sampling certificate, not the preceding separation results. If every finite pair was actually covered, completeness must be established separately rather than inferred from the word sample. The exact result, retaining the requested order, is ([[c4, a4, [[c4], [a4, b4]]]], Yes, 1, 3, No, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r03-hausdorff-space-0080/)

Hausdorff space · qf-mod42-hausdorff-sampled-boundary · candidate





## 81. Successful Pair Samples Do Not Certify Unchecked or Infinite Cases

Use finite unique labeled points=[a5, b5, c5] and the complete canonical topology=[[a5, b5, c5], [a5, b5], [a5, c5], [a5], [c5], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Supplied sampled_pairs=[[c5, a5]]. Give (sampled_pair_rows, all_sampled_separated, sample_count, total_pair_count, general_sample_certificate, boundary).

The fixed No withholds a general sampled-space or unchecked-pair certificate; it does not negate the separation flags. If all finite pairs were actually covered, that completeness must be stated separately.

View answer and explanation([[c5, a5, [[c5], [a5, b5]]]], Yes, 1, 3, No, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)

Audit each sampled pair and retain sample count and total finite unordered-pair count separately. The fixed No withholds a general sampling certificate, not the preceding separation results. If every finite pair was actually covered, completeness must be established separately rather than inferred from the word sample. The exact result, retaining the requested order, is ([[c5, a5, [[c5], [a5, b5]]]], Yes, 1, 3, No, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r03-hausdorff-space-0081/)

Hausdorff space · qf-mod42-hausdorff-sampled-boundary · candidate





## 82. Successful Pair Samples Do Not Certify Unchecked or Infinite Cases

Use finite unique labeled points=[a6, b6, c6] and the complete canonical topology=[[a6, b6, c6], [a6, b6], [a6, c6], [a6], [c6], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Supplied sampled_pairs=[[c6, a6]]. Give (sampled_pair_rows, all_sampled_separated, sample_count, total_pair_count, general_sample_certificate, boundary).

The fixed No withholds a general sampled-space or unchecked-pair certificate; it does not negate the separation flags. If all finite pairs were actually covered, that completeness must be stated separately.

View answer and explanation([[c6, a6, [[c6], [a6, b6]]]], Yes, 1, 3, No, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)

Audit each sampled pair and retain sample count and total finite unordered-pair count separately. The fixed No withholds a general sampling certificate, not the preceding separation results. If every finite pair was actually covered, completeness must be established separately rather than inferred from the word sample. The exact result, retaining the requested order, is ([[c6, a6, [[c6], [a6, b6]]]], Yes, 1, 3, No, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r03-hausdorff-space-0082/)

Hausdorff space · qf-mod42-hausdorff-sampled-boundary · candidate





## 83. Successful Pair Samples Do Not Certify Unchecked or Infinite Cases

Use finite unique labeled points=[a7, b7, c7] and the complete canonical topology=[[a7, b7, c7], [a7, b7], [a7, c7], [a7], [c7], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Supplied sampled_pairs=[[c7, a7]]. Give (sampled_pair_rows, all_sampled_separated, sample_count, total_pair_count, general_sample_certificate, boundary).

The fixed No withholds a general sampled-space or unchecked-pair certificate; it does not negate the separation flags. If all finite pairs were actually covered, that completeness must be stated separately.

View answer and explanation([[c7, a7, [[c7], [a7, b7]]]], Yes, 1, 3, No, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)

Audit each sampled pair and retain sample count and total finite unordered-pair count separately. The fixed No withholds a general sampling certificate, not the preceding separation results. If every finite pair was actually covered, completeness must be established separately rather than inferred from the word sample. The exact result, retaining the requested order, is ([[c7, a7, [[c7], [a7, b7]]]], Yes, 1, 3, No, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r03-hausdorff-space-0083/)

Hausdorff space · qf-mod42-hausdorff-sampled-boundary · candidate





## 84. Successful Pair Samples Do Not Certify Unchecked or Infinite Cases

Use finite unique labeled points=[a8, b8, c8] and the complete canonical topology=[[a8, b8, c8], [a8, b8], [a8, c8], [a8], [c8], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Supplied sampled_pairs=[[c8, a8]]. Give (sampled_pair_rows, all_sampled_separated, sample_count, total_pair_count, general_sample_certificate, boundary).

The fixed No withholds a general sampled-space or unchecked-pair certificate; it does not negate the separation flags. If all finite pairs were actually covered, that completeness must be stated separately.

View answer and explanation([[c8, a8, [[c8], [a8, b8]]]], Yes, 1, 3, No, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)

Audit each sampled pair and retain sample count and total finite unordered-pair count separately. The fixed No withholds a general sampling certificate, not the preceding separation results. If every finite pair was actually covered, completeness must be established separately rather than inferred from the word sample. The exact result, retaining the requested order, is ([[c8, a8, [[c8], [a8, b8]]]], Yes, 1, 3, No, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r03-hausdorff-space-0084/)

Hausdorff space · qf-mod42-hausdorff-sampled-boundary · candidate





## 85. Successful Pair Samples Do Not Certify Unchecked or Infinite Cases

Use finite unique labeled points=[a9, b9, c9] and the complete canonical topology=[[a9, b9, c9], [a9, b9], [a9, c9], [a9], [c9], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Supplied sampled_pairs=[[c9, a9]]. Give (sampled_pair_rows, all_sampled_separated, sample_count, total_pair_count, general_sample_certificate, boundary).

The fixed No withholds a general sampled-space or unchecked-pair certificate; it does not negate the separation flags. If all finite pairs were actually covered, that completeness must be stated separately.

View answer and explanation([[c9, a9, [[c9], [a9, b9]]]], Yes, 1, 3, No, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)

Audit each sampled pair and retain sample count and total finite unordered-pair count separately. The fixed No withholds a general sampling certificate, not the preceding separation results. If every finite pair was actually covered, completeness must be established separately rather than inferred from the word sample. The exact result, retaining the requested order, is ([[c9, a9, [[c9], [a9, b9]]]], Yes, 1, 3, No, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r03-hausdorff-space-0085/)

Hausdorff space · qf-mod42-hausdorff-sampled-boundary · candidate





## 86. Successful Pair Samples Do Not Certify Unchecked or Infinite Cases

Use finite unique labeled points=[a10, b10, c10] and the complete canonical topology=[[a10, b10, c10], [a10, b10], [a10, c10], [a10], [c10], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Supplied sampled_pairs=[[c10, a10]]. Give (sampled_pair_rows, all_sampled_separated, sample_count, total_pair_count, general_sample_certificate, boundary).

The fixed No withholds a general sampled-space or unchecked-pair certificate; it does not negate the separation flags. If all finite pairs were actually covered, that completeness must be stated separately.

View answer and explanation([[c10, a10, [[c10], [a10, b10]]]], Yes, 1, 3, No, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)

Audit each sampled pair and retain sample count and total finite unordered-pair count separately. The fixed No withholds a general sampling certificate, not the preceding separation results. If every finite pair was actually covered, completeness must be established separately rather than inferred from the word sample. The exact result, retaining the requested order, is ([[c10, a10, [[c10], [a10, b10]]]], Yes, 1, 3, No, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r03-hausdorff-space-0086/)

Hausdorff space · qf-mod42-hausdorff-sampled-boundary · candidate





## 87. Successful Pair Samples Do Not Certify Unchecked or Infinite Cases

Use finite unique labeled points=[a11, b11, c11] and the complete canonical topology=[[a11, b11, c11], [a11, b11], [a11, c11], [a11], [c11], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Supplied sampled_pairs=[[c11, a11]]. Give (sampled_pair_rows, all_sampled_separated, sample_count, total_pair_count, general_sample_certificate, boundary).

The fixed No withholds a general sampled-space or unchecked-pair certificate; it does not negate the separation flags. If all finite pairs were actually covered, that completeness must be stated separately.

View answer and explanation([[c11, a11, [[c11], [a11, b11]]]], Yes, 1, 3, No, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)

Audit each sampled pair and retain sample count and total finite unordered-pair count separately. The fixed No withholds a general sampling certificate, not the preceding separation results. If every finite pair was actually covered, completeness must be established separately rather than inferred from the word sample. The exact result, retaining the requested order, is ([[c11, a11, [[c11], [a11, b11]]]], Yes, 1, 3, No, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r03-hausdorff-space-0087/)

Hausdorff space · qf-mod42-hausdorff-sampled-boundary · candidate





## 88. Successful Pair Samples Do Not Certify Unchecked or Infinite Cases

Use finite unique labeled points=[a12, b12, c12] and the complete canonical topology=[[a12, b12, c12], [a12, b12], [a12, c12], [a12], [c12], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Supplied sampled_pairs=[[c12, a12]]. Give (sampled_pair_rows, all_sampled_separated, sample_count, total_pair_count, general_sample_certificate, boundary).

The fixed No withholds a general sampled-space or unchecked-pair certificate; it does not negate the separation flags. If all finite pairs were actually covered, that completeness must be stated separately.

View answer and explanation([[c12, a12, [[c12], [a12, b12]]]], Yes, 1, 3, No, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)

Audit each sampled pair and retain sample count and total finite unordered-pair count separately. The fixed No withholds a general sampling certificate, not the preceding separation results. If every finite pair was actually covered, completeness must be established separately rather than inferred from the word sample. The exact result, retaining the requested order, is ([[c12, a12, [[c12], [a12, b12]]]], Yes, 1, 3, No, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r03-hausdorff-space-0088/)

Hausdorff space · qf-mod42-hausdorff-sampled-boundary · candidate





## 89. Successful Pair Samples Do Not Certify Unchecked or Infinite Cases

Use finite unique labeled points=[a13, b13, c13] and the complete canonical topology=[[a13, b13, c13], [a13, b13], [a13, c13], [a13], [c13], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Supplied sampled_pairs=[[c13, a13]]. Give (sampled_pair_rows, all_sampled_separated, sample_count, total_pair_count, general_sample_certificate, boundary).

The fixed No withholds a general sampled-space or unchecked-pair certificate; it does not negate the separation flags. If all finite pairs were actually covered, that completeness must be stated separately.

View answer and explanation([[c13, a13, [[c13], [a13, b13]]]], Yes, 1, 3, No, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)

Audit each sampled pair and retain sample count and total finite unordered-pair count separately. The fixed No withholds a general sampling certificate, not the preceding separation results. If every finite pair was actually covered, completeness must be established separately rather than inferred from the word sample. The exact result, retaining the requested order, is ([[c13, a13, [[c13], [a13, b13]]]], Yes, 1, 3, No, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r03-hausdorff-space-0089/)

Hausdorff space · qf-mod42-hausdorff-sampled-boundary · candidate





## 90. Successful Pair Samples Do Not Certify Unchecked or Infinite Cases

Use finite unique labeled points=[a14, b14, c14] and the complete canonical topology=[[a14, b14, c14], [a14, b14], [a14, c14], [a14], [c14], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Supplied sampled_pairs=[[c14, a14]]. Give (sampled_pair_rows, all_sampled_separated, sample_count, total_pair_count, general_sample_certificate, boundary).

The fixed No withholds a general sampled-space or unchecked-pair certificate; it does not negate the separation flags. If all finite pairs were actually covered, that completeness must be stated separately.

View answer and explanation([[c14, a14, [[c14], [a14, b14]]]], Yes, 1, 3, No, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)

Audit each sampled pair and retain sample count and total finite unordered-pair count separately. The fixed No withholds a general sampling certificate, not the preceding separation results. If every finite pair was actually covered, completeness must be established separately rather than inferred from the word sample. The exact result, retaining the requested order, is ([[c14, a14, [[c14], [a14, b14]]]], Yes, 1, 3, No, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r03-hausdorff-space-0090/)

Hausdorff space · qf-mod42-hausdorff-sampled-boundary · candidate





## 91. Successful Pair Samples Do Not Certify Unchecked or Infinite Cases

Use finite unique labeled points=[a15, b15, c15] and the complete canonical topology=[[a15, b15, c15], [a15, b15], [a15, c15], [a15], [c15], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Supplied sampled_pairs=[[c15, a15]]. Give (sampled_pair_rows, all_sampled_separated, sample_count, total_pair_count, general_sample_certificate, boundary).

The fixed No withholds a general sampled-space or unchecked-pair certificate; it does not negate the separation flags. If all finite pairs were actually covered, that completeness must be stated separately.

View answer and explanation([[c15, a15, [[c15], [a15, b15]]]], Yes, 1, 3, No, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)

Audit each sampled pair and retain sample count and total finite unordered-pair count separately. The fixed No withholds a general sampling certificate, not the preceding separation results. If every finite pair was actually covered, completeness must be established separately rather than inferred from the word sample. The exact result, retaining the requested order, is ([[c15, a15, [[c15], [a15, b15]]]], Yes, 1, 3, No, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r03-hausdorff-space-0091/)

Hausdorff space · qf-mod42-hausdorff-sampled-boundary · candidate





## 92. Successful Pair Samples Do Not Certify Unchecked or Infinite Cases

Use finite unique labeled points=[a16, b16, c16] and the complete canonical topology=[[a16, b16, c16], [a16, b16], [a16, c16], [a16], [c16], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Supplied sampled_pairs=[[c16, a16]]. Give (sampled_pair_rows, all_sampled_separated, sample_count, total_pair_count, general_sample_certificate, boundary).

The fixed No withholds a general sampled-space or unchecked-pair certificate; it does not negate the separation flags. If all finite pairs were actually covered, that completeness must be stated separately.

View answer and explanation([[c16, a16, [[c16], [a16, b16]]]], Yes, 1, 3, No, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)

Audit each sampled pair and retain sample count and total finite unordered-pair count separately. The fixed No withholds a general sampling certificate, not the preceding separation results. If every finite pair was actually covered, completeness must be established separately rather than inferred from the word sample. The exact result, retaining the requested order, is ([[c16, a16, [[c16], [a16, b16]]]], Yes, 1, 3, No, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r03-hausdorff-space-0092/)

Hausdorff space · qf-mod42-hausdorff-sampled-boundary · candidate





## 93. Successful Pair Samples Do Not Certify Unchecked or Infinite Cases

Use finite unique labeled points=[a17, b17, c17] and the complete canonical topology=[[a17, b17, c17], [a17, b17], [a17, c17], [a17], [c17], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Supplied sampled_pairs=[[c17, a17]]. Give (sampled_pair_rows, all_sampled_separated, sample_count, total_pair_count, general_sample_certificate, boundary).

The fixed No withholds a general sampled-space or unchecked-pair certificate; it does not negate the separation flags. If all finite pairs were actually covered, that completeness must be stated separately.

View answer and explanation([[c17, a17, [[c17], [a17, b17]]]], Yes, 1, 3, No, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)

Audit each sampled pair and retain sample count and total finite unordered-pair count separately. The fixed No withholds a general sampling certificate, not the preceding separation results. If every finite pair was actually covered, completeness must be established separately rather than inferred from the word sample. The exact result, retaining the requested order, is ([[c17, a17, [[c17], [a17, b17]]]], Yes, 1, 3, No, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r03-hausdorff-space-0093/)

Hausdorff space · qf-mod42-hausdorff-sampled-boundary · candidate





## 94. Successful Pair Samples Do Not Certify Unchecked or Infinite Cases

Use finite unique labeled points=[a18, b18, c18] and the complete canonical topology=[[a18, b18, c18], [a18, b18], [a18, c18], [a18], [c18], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Supplied sampled_pairs=[[c18, a18]]. Give (sampled_pair_rows, all_sampled_separated, sample_count, total_pair_count, general_sample_certificate, boundary).

The fixed No withholds a general sampled-space or unchecked-pair certificate; it does not negate the separation flags. If all finite pairs were actually covered, that completeness must be stated separately.

View answer and explanation([[c18, a18, [[c18], [a18, b18]]]], Yes, 1, 3, No, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)

Audit each sampled pair and retain sample count and total finite unordered-pair count separately. The fixed No withholds a general sampling certificate, not the preceding separation results. If every finite pair was actually covered, completeness must be established separately rather than inferred from the word sample. The exact result, retaining the requested order, is ([[c18, a18, [[c18], [a18, b18]]]], Yes, 1, 3, No, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r03-hausdorff-space-0094/)

Hausdorff space · qf-mod42-hausdorff-sampled-boundary · candidate





## 95. Successful Pair Samples Do Not Certify Unchecked or Infinite Cases

Use finite unique labeled points=[a19, b19, c19] and the complete canonical topology=[[a19, b19, c19], [a19, b19], [a19, c19], [a19], [c19], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Supplied sampled_pairs=[[c19, a19]]. Give (sampled_pair_rows, all_sampled_separated, sample_count, total_pair_count, general_sample_certificate, boundary).

The fixed No withholds a general sampled-space or unchecked-pair certificate; it does not negate the separation flags. If all finite pairs were actually covered, that completeness must be stated separately.

View answer and explanation([[c19, a19, [[c19], [a19, b19]]]], Yes, 1, 3, No, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)

Audit each sampled pair and retain sample count and total finite unordered-pair count separately. The fixed No withholds a general sampling certificate, not the preceding separation results. If every finite pair was actually covered, completeness must be established separately rather than inferred from the word sample. The exact result, retaining the requested order, is ([[c19, a19, [[c19], [a19, b19]]]], Yes, 1, 3, No, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r03-hausdorff-space-0095/)

Hausdorff space · qf-mod42-hausdorff-sampled-boundary · candidate





## 96. Successful Pair Samples Do Not Certify Unchecked or Infinite Cases

Use finite unique labeled points=[a20, b20, c20] and the complete canonical topology=[[a20, b20, c20], [a20, b20], [a20, c20], [a20], [c20], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Supplied sampled_pairs=[[c20, a20]]. Give (sampled_pair_rows, all_sampled_separated, sample_count, total_pair_count, general_sample_certificate, boundary).

The fixed No withholds a general sampled-space or unchecked-pair certificate; it does not negate the separation flags. If all finite pairs were actually covered, that completeness must be stated separately.

View answer and explanation([[c20, a20, [[c20], [a20, b20]]]], Yes, 1, 3, No, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)

Audit each sampled pair and retain sample count and total finite unordered-pair count separately. The fixed No withholds a general sampling certificate, not the preceding separation results. If every finite pair was actually covered, completeness must be established separately rather than inferred from the word sample. The exact result, retaining the requested order, is ([[c20, a20, [[c20], [a20, b20]]]], Yes, 1, 3, No, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r03-hausdorff-space-0096/)

Hausdorff space · qf-mod42-hausdorff-sampled-boundary · candidate





## 97. Successful Pair Samples Do Not Certify Unchecked or Infinite Cases

Use finite unique labeled points=[a21, b21, c21] and the complete canonical topology=[[a21, b21, c21], [a21, b21], [a21, c21], [a21], [c21], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Supplied sampled_pairs=[[c21, a21]]. Give (sampled_pair_rows, all_sampled_separated, sample_count, total_pair_count, general_sample_certificate, boundary).

The fixed No withholds a general sampled-space or unchecked-pair certificate; it does not negate the separation flags. If all finite pairs were actually covered, that completeness must be stated separately.

View answer and explanation([[c21, a21, [[c21], [a21, b21]]]], Yes, 1, 3, No, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)

Audit each sampled pair and retain sample count and total finite unordered-pair count separately. The fixed No withholds a general sampling certificate, not the preceding separation results. If every finite pair was actually covered, completeness must be established separately rather than inferred from the word sample. The exact result, retaining the requested order, is ([[c21, a21, [[c21], [a21, b21]]]], Yes, 1, 3, No, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r03-hausdorff-space-0097/)

Hausdorff space · qf-mod42-hausdorff-sampled-boundary · candidate





## 98. Successful Pair Samples Do Not Certify Unchecked or Infinite Cases

Use finite unique labeled points=[a22, b22, c22] and the complete canonical topology=[[a22, b22, c22], [a22, b22], [a22, c22], [a22], [c22], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Supplied sampled_pairs=[[c22, a22]]. Give (sampled_pair_rows, all_sampled_separated, sample_count, total_pair_count, general_sample_certificate, boundary).

The fixed No withholds a general sampled-space or unchecked-pair certificate; it does not negate the separation flags. If all finite pairs were actually covered, that completeness must be stated separately.

View answer and explanation([[c22, a22, [[c22], [a22, b22]]]], Yes, 1, 3, No, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)

Audit each sampled pair and retain sample count and total finite unordered-pair count separately. The fixed No withholds a general sampling certificate, not the preceding separation results. If every finite pair was actually covered, completeness must be established separately rather than inferred from the word sample. The exact result, retaining the requested order, is ([[c22, a22, [[c22], [a22, b22]]]], Yes, 1, 3, No, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r03-hausdorff-space-0098/)

Hausdorff space · qf-mod42-hausdorff-sampled-boundary · candidate





## 99. Successful Pair Samples Do Not Certify Unchecked or Infinite Cases

Use finite unique labeled points=[a23, b23, c23] and the complete canonical topology=[[a23, b23, c23], [a23, b23], [a23, c23], [a23], [c23], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Supplied sampled_pairs=[[c23, a23]]. Give (sampled_pair_rows, all_sampled_separated, sample_count, total_pair_count, general_sample_certificate, boundary).

The fixed No withholds a general sampled-space or unchecked-pair certificate; it does not negate the separation flags. If all finite pairs were actually covered, that completeness must be stated separately.

View answer and explanation([[c23, a23, [[c23], [a23, b23]]]], Yes, 1, 3, No, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)

Audit each sampled pair and retain sample count and total finite unordered-pair count separately. The fixed No withholds a general sampling certificate, not the preceding separation results. If every finite pair was actually covered, completeness must be established separately rather than inferred from the word sample. The exact result, retaining the requested order, is ([[c23, a23, [[c23], [a23, b23]]]], Yes, 1, 3, No, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r03-hausdorff-space-0099/)

Hausdorff space · qf-mod42-hausdorff-sampled-boundary · candidate





## 100. Successful Pair Samples Do Not Certify Unchecked or Infinite Cases

Use finite unique labeled points=[a24, b24, c24] and the complete canonical topology=[[a24, b24, c24], [a24, b24], [a24, c24], [a24], [c24], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Supplied sampled_pairs=[[c24, a24]]. Give (sampled_pair_rows, all_sampled_separated, sample_count, total_pair_count, general_sample_certificate, boundary).

The fixed No withholds a general sampled-space or unchecked-pair certificate; it does not negate the separation flags. If all finite pairs were actually covered, that completeness must be stated separately.

View answer and explanation([[c24, a24, [[c24], [a24, b24]]]], Yes, 1, 3, No, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)

Audit each sampled pair and retain sample count and total finite unordered-pair count separately. The fixed No withholds a general sampling certificate, not the preceding separation results. If every finite pair was actually covered, completeness must be established separately rather than inferred from the word sample. The exact result, retaining the requested order, is ([[c24, a24, [[c24], [a24, b24]]]], Yes, 1, 3, No, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r03-hausdorff-space-0100/)

Hausdorff space · qf-mod42-hausdorff-sampled-boundary · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261111-r03-hausdorff-space-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261014-advance-20261111-r03-hausdorff-space-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261111-r03-hausdorff-space-foundations/validation-report.json)