Complete Point-Pair Separators, Vacuous Cases and Sample Nonconclusions

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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.

Reviewed step-by-step derivation

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.

Reviewed step-by-step derivation

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.

Reviewed step-by-step derivation

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.

Reviewed step-by-step derivation

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.

Reviewed step-by-step derivation

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.

Reviewed step-by-step derivation

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.

Reviewed step-by-step derivation

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.

Reviewed step-by-step derivation

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.

Reviewed step-by-step derivation

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.

Reviewed step-by-step derivation

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.

Reviewed step-by-step derivation

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.

Reviewed step-by-step derivation

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.

Reviewed step-by-step derivation

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.

Reviewed step-by-step derivation

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.

Reviewed step-by-step derivation

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.

Reviewed step-by-step derivation

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.

Reviewed step-by-step derivation

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.

Reviewed step-by-step derivation

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.

Reviewed step-by-step derivation

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.

Reviewed step-by-step derivation

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.

Reviewed step-by-step derivation

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.

Reviewed step-by-step derivation

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.

Reviewed step-by-step derivation

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.

Reviewed step-by-step derivation

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.

Reviewed step-by-step derivation

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