DefinitionDefinition: Literature-sourcedProof: Not applicablejudge 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.

Fréchet–Urysohn spaces and sequential spaces

Definition

A topological space $X$ is Fréchet–Urysohn if, whenever $p \in \overline{A}$ , there is a sequence in $A$ converging to $p$ . Equivalently, $seqcl (A) = \overline{A}$ for every $A \subseteq X$ , since sequential closure is always contained in closure (The sequential closure is contained in the closure, continuity implies sequential continuity, and sequential limits need not be unique).

A subset $C \subseteq X$ is sequentially closed if every sequence in $C$ that converges in $X$ has its limit in $C$ . The space is sequential if every sequentially closed subset is closed. Equivalently, $seqcl (A) = A$ implies $\overline{A} = A$ for every $A \subseteq X$ .

Depends on

Used by

Dependency tree · two levels

15 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

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