Completeness and related properties of the graph topology on function spaces

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 properti...

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Bibliographic Details
Published inarXiv.org
Main Authors Holá, Lubica, Zsilinszky, László
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 24.04.2013
Subjects
Completeness
Continuity (mathematics)
Function space
Properties (attributes)
Topology
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Summary: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.
ISSN:2331-8422