Completeness and related properties of the graph topology on function spaces

Lubica Holá; László Zsilinszky

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arXiv · 1304.6628

Completeness and related properties of the graph topology on function spaces

Lubica Holá·László Zsilinszky

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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.

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BibTeX RIS

Lubica Holá, László Zsilinszky. 2013-04-24. Completeness and related properties of the graph topology on function spaces. https://arxiv.org/abs/1304.6628

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