Completeness and related properties of the graph topology on function spaces

TL;DR

This paper investigates the completeness and related topological properties of the graph topology on spaces of continuous functions, revealing conditions under which various completeness properties coincide and exploring their behavior in different topological contexts.

Contribution

It characterizes when all intermediate completeness properties coincide for the graph topology on $C(X)$ and analyzes related properties like pseudocompleteness and countability.

Findings

All completeness properties between complete metrizability and hereditary Baireness coincide if and only if $X$ is countably compact.

The graph topology is $ ext{α}$-favorable in the strong Choquet game regardless of $X$.

Analogous results are obtained for the fine topology on $C(X)$.

Abstract

The graph topology $\tau_{\Gamma}$ is the topology on the space $C(X)$ of all continuous functions defined on a Tychonoff space $X$ inherited from the Vietoris topology on $X\times \mathbb R$ after identifying continuous functions with their graphs. It is shown that all completeness properties between complete metrizability and hereditary Baireness coincide for the graph topology if and only if $X$ is countably compact; however, the graph topology is $\alpha$-favorable in the strong Choquet game, regardless of $X$. Analogous results are obtained for the fine topology on $C(X)$. Pseudocompleteness, along with properties related to 1st and 2nd countability of $(C(X),\tau_{\Gamma})$ are also investigated.