CounterexampleConstruction: AI-adaptedVerification: AI-generatedprecheck passjudge pass (z-ai/glm-5.2)audited 2026-07-31

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Regular and normal do not imply $T_{1}$ under the library's conventions

Statement refuted

Every regular space, and every normal space, is $T_{1}$ .

Facts & Assumptions

Given: A two-point set $X$ with its indiscrete topology.

[F1]

The indiscrete topology has only $\emptyset$ and $X$ as open sets, hence only $\emptyset$ and $X$ as closed sets (The discrete, indiscrete, cofinite, cocountable, particular-point and Sierpinski topologies).

[F2]

Regularity separates a point from a closed set that misses it; normality separates disjoint closed sets; $T_{1}$ separates each distinct point from the other by an open set (Regular spaces and $T_{3}$ spaces, with the source disagreement over whether regularity includes $T_{1}$ stated explicitly, Normal spaces and $T_{4}$ spaces, with the source disagreement over whether normality includes $T_{1}$ stated explicitly, $T_{0}$ (Kolmogorov) and $T_{1}$ (Frechet) spaces).

Counterexample

technique · direct

1.1

In this topology the only closed set missing a point is $\emptyset$ , and it is separated from that point by $X$ and $\emptyset$ ; thus $X$ is regular.

F1F2

1.2

Every disjoint pair of closed sets has an empty member, so the same two open sets show that $X$ is normal.

F1F2

2.1

No open set contains one point while missing the other, so $X$ is not $T_{1}$ . This refutes both implications.

F1F2step 1.1step 1.2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

18 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

S. Willard, General Topology, separation axioms (standard reference, not scraped)
Sierpinski space (Wikipedia) (standard reference, not scraped)
Separation axiom (Wikipedia) (standard reference, not scraped)

Regular and normal do not imply T1T1​ under the library's conventions