Separation Axioms: the Hierarchy

About this page

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.

✓ 17 results · all verified · 16 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 1 not AI-judged were verified by owner audit (typically over a confirmed judge false positive), not failures.

2 · Logical flowchart

3 · Items

Definition

T₀ (Kolmogorov) and T₁ (Frechet) spaces

Theorem

A space is T₁ if and only if every singleton is closed, if and only if every finite subset is closed, if and only if its topology contains the cofinite topology

Definition

A space is regular if and only if every point has a neighbourhood base of closed neighbourhoods, if and only if x ∈ U open gives an open V with x ∈ V ⊆ V̄ ⊆ U

Definition

Assuming countable choice, every perfectly normal space is completely normal: separated sets in a normal space whose open sets are all F_σ can be separated by disjoint open sets

Theorem

In a metric space any two separated sets have disjoint open neighbourhoods, so every metrizable space is completely normal

Theorem

In a metric space every closed set is a zero set and a G_δ, and the distance function separates a point from a closed set, so every metrizable space is Tychonoff and perfectly normal

Theorem

The implications proved on this page: perfectly normal gives completely normal under countable choice, and completely normal gives normal; normal with T₁ gives T₃; completely regular gives regular; regular with T₁ gives Urysohn, hence Hausdorff, hence T₁, hence T₀; and metrizable gives every one of them

Definition

The order topology on an ordinal, with the half-open intervals (α, β] and the initial segments [0, β] as a basis

Lemma

Every ordinal with its order topology has a basis of clopen sets, and is T₁, Hausdorff and regular

Lemma

The K-topology on ℝ, generated by the open intervals together with their complements of K = {1/(n+1) : n ∈ ℕ}, is T₁ and Hausdorff but not regular

Remark

Conventions on this page, and the one implication of the classical chain that is not available at this point in the reading order

False statement

Separation Axioms: the Hierarchy

3 · Items

T₀ (Kolmogorov) and T₁ (Frechet) spaces

A space is T₁ if and only if every singleton is closed, if and only if every finite subset is closed, if and only if its topology contains the cofinite topology

Separated sets: Ā ∩ B = A ∩ B̄ = ∅

G_δ and F_σ subsets of a topological space, agreeing with the real-line notion

Regular spaces and T₃ spaces, with the source disagreement over whether regularity includes T₁ stated explicitly

A space is regular if and only if every point has a neighbourhood base of closed neighbourhoods, if and only if x ∈ U open gives an open V with x ∈ V ⊆ V̄ ⊆ U

Urysohn (T_21/2) space: distinct points have neighbourhoods with disjoint closures

Every Urysohn space is Hausdorff, every Hausdorff space is T₁ and hence T₀, and every regular T₁ space is Urysohn

Zero sets and cozero sets of continuous real-valued functions

Completely regular spaces and Tychonoff (T_31/2) spaces

Every completely regular space is regular, and every Tychonoff space is T₃

Normal spaces and T₄ spaces, with the source disagreement over whether normality includes T₁ stated explicitly

A space is normal if and only if every closed A inside an open U admits an open V with A ⊆ V ⊆ V̄ ⊆ U

A normal T₁ space is regular, hence T₃, hence Urysohn, Hausdorff, T₁ and T₀

Completely normal (T₅) and perfectly normal (T₆) spaces

Every completely normal space is normal, and every perfectly normal space is normal

Assuming countable choice, every perfectly normal space is completely normal: separated sets in a normal space whose open sets are all F_σ can be separated by disjoint open sets

In a metric space any two separated sets have disjoint open neighbourhoods, so every metrizable space is completely normal

In a metric space every closed set is a zero set and a G_δ, and the distance function separates a point from a closed set, so every metrizable space is Tychonoff and perfectly normal

The order topology on an ordinal, with the half-open intervals (α, β] and the initial segments [0, β] as a basis

Every ordinal with its order topology has a basis of clopen sets, and is T₁, Hausdorff and regular

The K-topology on ℝ, generated by the open intervals together with their complements of K = {1/(n+1) : n ∈ ℕ}, is T₁ and Hausdorff but not regular

Conventions on this page, and the one implication of the classical chain that is not available at this point in the reading order

FALSE: every T₁ space is Hausdorff

FALSE: every normal space is Hausdorff, so the T₁ hypothesis in T₄ is redundant

FALSE: every Hausdorff space is regular

FALSE: a space in which every sequence has at most one limit is Hausdorff