TheoremStatement: AI-adaptedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (z-ai/glm-5.2)audited 2026-07-31

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Assuming countable choice, every first countable space is Fréchet–Urysohn; in ZF every Fréchet–Urysohn space is sequential

Statement

Assume countable choice. Every first countable space is Fréchet–Urysohn. In ZF, every Fréchet–Urysohn space is sequential.

Facts & Assumptions

Given: A topological space $X$ .

[L1]

Under countable choice, first countability gives $\operatorname{seqcl}(A)=\overline A$ for every $A\subseteq X$ (Assuming Countable Choice, in a first countable space sequential closure equals closure and sequential continuity at a point equals continuity there, The Axiom of Countable Choice ( $\mathrm{AC}_\omega$ )).

[L2]

$A\subseteq\operatorname{seqcl}(A)\subseteq\overline A$ for every $A$ (The sequential closure is contained in the closure, continuity implies sequential continuity, and sequential limits need not be unique).

[A1]

A space is Fréchet--Urysohn when $\operatorname{seqcl}(A)=\overline A$ for every subset $A$ , and it is sequential when every sequentially closed subset is closed (Fréchet–Urysohn spaces and sequential spaces).

Proof

technique · direct

1.1

Under countable choice, [L1] is exactly the defining equality for a first countable space to be Fréchet–Urysohn.

L1A1

1.2

Now suppose $X$ is Fréchet–Urysohn and $C$ is sequentially closed. Then $\operatorname{seqcl}(C)=C$ , because the constant sequence gives $C\subseteq\operatorname{seqcl}(C)$ and sequential closedness gives the reverse inclusion.

2.1

Fréchet–Urysohnness gives $\overline C=\operatorname{seqcl}(C)=C$ , so $C$ is closed. Therefore $X$ is sequential.

step 1.2A1∎

Depends on

Used by

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Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 45 results over 18 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

Sequential space (Wikipedia) (standard reference, not scraped)
Fréchet–Urysohn space (Wikipedia) (standard reference, not scraped)
First-countable space (Wikipedia) (standard reference, not scraped)