TheoremStatement: AI-adaptedProof: 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.

$T_{0}$ , $T_{1}$ , $T_{2}$ , regularity, $T_{3}$ , complete regularity, and Tychonoffness are productive

Statement

Each of $T_{0}$ , $T_{1}$ , Hausdorffness ( $T_{2}$ ), regularity, $T_{3}$ , complete regularity, and Tychonoffness is productive.

Facts & Assumptions

Given: A family of spaces having one of the listed separation properties.

[L1]

Arbitrary products preserve $T_{0}$ , $T_{1}$ , Hausdorffness, regularity, and complete regularity as stated in the three preceding lemmas (Arbitrary products preserve $T_{0}$ , $T_{1}$ , and Hausdorffness, Arbitrary products of regular spaces are regular, Arbitrary products of completely regular spaces are completely regular).

[F1]

$T_{3}$ is regular plus $T_{1}$ , and Tychonoff is completely regular plus $T_{1}$ (Regular spaces and $T_{3}$ spaces, with the source disagreement over whether regularity includes $T_{1}$ stated explicitly, Completely regular spaces and Tychonoff ( $T_{3 \frac{1}{2}}$ ) spaces).

Proof

technique · direct

1.1

The assertions for $T_{0}$ , $T_{1}$ , $T_{2}$ , regularity, and complete regularity are [L1].

1.2

A product of $T_{3}$ spaces is regular and $T_{1}$ by [L1], hence is $T_{3}$ .

F1L1

1.3

A product of Tychonoff spaces is completely regular and $T_{1}$ by [L1], hence is Tychonoff.

F1L1

2.1

Therefore every property in the statement is productive.

step 1.1step 1.2step 1.3∎

Depends on

Used by

Dependency tree · two levels

21 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)
Separation axiom (Wikipedia) (standard reference, not scraped)

T0T0​, T1T1​, T2T2​, regularity, T3T3​, complete regularity, and Tychonoffness are productive