Alphabeta Math

Separation Axioms: the Hierarchy: Examples and Counterexamples

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✓ 7 results · all verified · 7 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; all 7 also cleared it.
1 · Prerequisites
Real Analysis
Set Theory
Topology
2 · Logical flowchart

3 · Items

Example

Sierpinski space is T₀ and normal but neither T₁ nor regular: normality without T₁ implies nothing

Example

The particular-point topology is T₀, it is not T₁ and not regular once the set has at least two points, and it is not normal once the set has at least three

Example

The cofinite topology on an infinite set is T₁ but neither Hausdorff nor regular nor normal

Example

The cocountable topology on ℝ is T₁, has unique sequential limits, and is neither Hausdorff nor regular nor normal

Example

A discrete space satisfies every axiom in the chain; an indiscrete space with two points is regular, completely regular, normal, completely normal and perfectly normal, and fails T₀

Example

Every nonempty closed subset A of ℝ is the zero set of x ↦ d(x, A) and the intersection of the open sets {x : d(x,A) < 1/(n+1)}, worked for [0,1] and for {0}

Example

ω + 1 as a convergent sequence together with its limit, and, assuming countable choice, [0, ω₁), in which every sequence lies inside an at most countable initial segment

Counterexample

Refuted, assuming countable choice: every Hausdorff space built from ordinal spaces is normal. The deleted Tychonoff plank ((ω₁ + 1) × (ω + 1)) ∖ {(ω₁, ω)} is Hausdorff and not normal

Sources