CorollaryStatement: AI-generatedProof: 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.

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 · next 3 levels

Direct dependencies and their dependencies through the next three levels: 34 results over 14 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

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

Complete normality, and hence T5T_5T5​, is hereditary