LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck 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.

Every closed subspace of a normal space is normal

Statement

Normality is closed-hereditary: every closed subspace of a normal space is normal.

Facts & Assumptions

Given: A normal space $X$ , a closed subspace $S \subseteq X$ , and disjoint closed subsets $A, B$ of $S$ .

[F1]

A set closed in a subspace is the trace of an ambient closed set; if the subspace is closed, it is itself ambient closed (Subspace topology: the traces of the open sets, its closed sets and its bases, the continuity of the inclusion, and the characteristic property of a map into a subspace).

[F2]

Normality separates disjoint closed subsets by disjoint open sets (Normal spaces and $T_{4}$ spaces, with the source disagreement over whether normality includes $T_{1}$ stated explicitly).

Proof

technique · direct

1.1

Write $A = S \cap C$ and $B = S \cap D$ for closed $C, D \subseteq X$ . Then $A$ and $B$ are closed in $X$ , because $S$ is closed.

2.1

The sets $A, B$ are disjoint closed subsets of the normal space $X$ , so choose disjoint ambient open sets $U, V$ with $A \subseteq U$ and $B \subseteq V$ .

F2step 1.1

3.1

Their traces $U \cap S$ and $V \cap S$ are disjoint open subsets of $S$ containing $A, B$ , so $S$ is normal.

F1step 2.1∎

Depends on

Used by

Dependency tree · two levels

12 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

J. P. May, An Outline Summary of Basic Point Set Topology, §6 (standard reference, not scraped)
Normal space (Wikipedia) (standard reference, not scraped)