RemarkRemark: AI-adaptedProof: Not applicablePipeline-generated‡ sources checked 2026-09-22‡ not proved here

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.

‡ Recorded, not proved here. This statement is included so the library can refer to it honestly, with a citation to the literature. It is not proved anywhere in this library: the track that would prove it has not been developed here yet.

Gerlits Nagy remains selection principle theory

Statement

For a Tychonoff space $X$ the following are equivalent: $C_{p} (X)$ is Fréchet–Urysohn; $C_{p} (X)$ is sequential; $C_{p} (X)$ is a $k$ -space; and $X$ has the $γ$ -property, in the sense that every open $ω$ -cover of $X$ contains a $γ$ -subcover. Here an $ω$ -cover is an open cover of $X$ not containing $X$ as a member such that every finite subset of $X$ is contained in some member, and a $γ$ -cover is an infinite open cover such that every point of $X$ belongs to all but finitely many members.

This result is recorded, not proved here, and it is not used anywhere in this pair; the selection-principle theory is not part of the Gelfand programme on this page. The catalogue target is Gerlits-Nagy theorem: $C_{p} (X)$ is Frechet-Urysohn exactly for gamma-spaces ‡.

Remarks

Orientation only. The remark exists so that the four-way equivalence is not silently attributed to the Gelfand-theoretic machinery of this page.
Open obligation for the catalogue. Obtaining and inspecting the complete original proof, or an equivalent complete treatment, and aligning the selection-principle conventions is a repair obligation on the catalogue target.

Used by

Nothing in the library uses this result yet.

Dependency tree · 0 levels

Nothing. This result depends on no other item in the library.

Sources

S. Gabriyelyan and A. Osipov, Topological properties of some function spaces (2020) — Theorem 1.3, p. 2, and §2.1 definitions, p. 5; the paper states the equivalence and cites the original proof (standard reference, not scraped)