False statementConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck 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.

Refuted: every paracompact space is normal

Statement

Every paracompact space is normal.

Facts & Assumptions

Given: The three-point set $X=\{a,b,c\}$ with topology $\{\varnothing,\{c\},\{a,c\},\{b,c\},X\}$ .

[F1]

A compact space is paracompact (Every compact space is paracompact).

[F2]

Normality separates every disjoint pair of closed sets by disjoint open sets (Normal spaces and $T_4$ spaces, with the source disagreement over whether normality includes $T_1$ stated explicitly).

Refutation

technique · direct

1.1

The displayed family is a topology, and $X$ is compact because every open cover of this finite set already has a finite subcover.

construct

2.1

Its closed sets include $\{a\}$ and $\{b\}$ , while every open set containing $a$ contains $c$ and every open set containing $b$ contains $c$ .

step 1.1

3.1

Thus the disjoint closed sets $\{a\}$ and $\{b\}$ have no disjoint open neighbourhoods, so $X$ is not normal by [F2].

F2step 2.1

4.1

By [F1] the compact space $X$ is paracompact, and step 3.1 refutes the displayed assertion.

F1step 1.1step 3.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 26 results over 11 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

J. Robbin, Partitions of Unity (standard reference, not scraped)
Paracompact space (Wikipedia) (standard reference, not scraped)
R. Gardner, Notes on Munkres Section 41: Paracompactness (East Tennessee State University) (standard reference, not scraped)