CorollaryStatement: AI-generatedProof: 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.

Complete normality, and hence $T_{5}$ , is hereditary

Statement

Complete normality is hereditary. Consequently $T_{5}$ is hereditary.

Facts & Assumptions

Given: A subspace of a completely normal, respectively $T_{5}$ , space.

[L1]

A completely normal space is exactly a space all of whose subspaces are normal (A space is completely normal if and only if every subspace is normal).

[L2]

$T_{1}$ is hereditary ( $T_{0}$ , $T_{1}$ , and Hausdorffness are hereditary).

[F1]

$T_{5}$ means completely normal plus $T_{1}$ (Completely normal ( $T_{5}$ ) and perfectly normal ( $T_{6}$ ) spaces).

Proof

technique · direct

1.1

If $S \subseteq X$ and $X$ is completely normal, every subspace of $S$ is also a subspace of $X$ and hence is normal by [L1]. Therefore [L1] applied to $S$ makes $S$ completely normal.

2.1

If $X$ is $T_{5}$ , step 1.1 gives complete normality of $S$ and [L2] gives $T_{1}$ of $S$ , so $S$ is $T_{5}$ .

F1L2step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

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

S. Willard, General Topology, §15 (standard reference, not scraped)

Complete normality, and hence T5T5​, is hereditary