Alphabeta Math

Hereditary and Productive Behaviour of the Separation Axioms

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✓ 22 results · all verified · 19 also independently AI-judged
Every result on this page is machine-checked by a proof checker and read in full and owner-audited; the judge is an additional, independent cross-model AI review of the proofs. The 3 not AI-judged were verified by owner audit (typically over a confirmed judge false positive), not failures.
1 · Prerequisites
Real Analysis
Set Theory
Topology
2 · Logical flowchart

3 · Items

Lemma

T₀, T₁, and Hausdorffness are hereditary

Lemma

Regularity is hereditary, without a hidden T₁ hypothesis

Lemma

Complete regularity is hereditary, without a hidden T₁ hypothesis

Theorem

T₀, T₁, T₂, regularity, T₃, complete regularity, and Tychonoffness are hereditary

Lemma

Arbitrary products preserve T₀, T₁, and Hausdorffness

Lemma

Arbitrary products of regular spaces are regular

Lemma

Finite pointwise minima of continuous maps to [0,1] are continuous

Lemma

Arbitrary products of completely regular spaces are completely regular

Theorem

T₀, T₁, T₂, regularity, T₃, complete regularity, and Tychonoffness are productive

Lemma

Every closed subspace of a normal space is normal

Theorem

A space is completely normal if and only if every subspace is normal

Corollary

Complete normality, and hence T₅, is hereditary

Corollary

Assuming countable choice, perfect normality, and hence T₆, is hereditary

Lemma

Every regular Lindelöf space is normal

Definition

The lower-limit topology on ℝ, with the half-open intervals [a,b) as a basis

Lemma

The lower-limit line has a clopen basis, is regular, and is Lindelöf under countable choice

Corollary

Assuming countable choice, the lower-limit line is normal

Lemma

Jones's bound: under choice, a closed discrete subspace of a normal space cannot have more subsets than a dense set has subsets

Lemma

The lower-limit plane has a countable dense set and a closed discrete antidiagonal of size |ℝ|

Theorem

Assuming choice, normality is not productive: the normal lower-limit line has a nonnormal square

Lemma

Assuming countable choice, the deleted Tychonoff plank is a regular nonnormal open subspace of a compact Hausdorff normal space

Theorem

Assuming countable choice, normality is not hereditary, even to open regular subspaces

False statement

Assuming countable choice, refuted: every regular space is normal

Remark

Preservation ledger for the separation axioms, with T₁ conventions kept explicit

Sources