Topological properties of some function spaces

Saak Gabriyelyan, Alexander V. Osipov

Research output: Contribution to journalArticlepeer-review

1 Scopus citations

Abstract

Let Y be a metrizable space containing at least two points, and let X be a YI-Tychonoff space for some ideal I of compact sets of X. Denote by CI(X,Y) the space of continuous functions from X to Y endowed with the I-open topology. We prove that CI(X,Y) is Fréchet–Urysohn iff X has the property γI. We characterize zero-dimensional Tychonoff spaces X for which the space CI(X,2) is sequential. Extending the classical theorems of Gerlits, Nagy and Pytkeev we show that if Y is not compact, then Cp(X,Y) is Fréchet–Urysohn iff it is sequential iff it is a k-space iff X has the property γ. An analogous result is obtained for the space of bounded continuous functions taking values in a metrizable locally convex space. Denote by B1(X,Y) and B(X,Y) the space of Baire one functions and the space of all Baire functions from X to Y, respectively. If H is a subspace of B(X,Y) containing B1(X,Y), then H is metrizable iff it is a σ-space iff it has countable cs-character iff X is countable. If additionally Y is not compact, then H is Fréchet–Urysohn iff it is sequential iff it is a k-space iff it has countable tightness iff X0 has the property γ, where X0 is the space X with the Baire topology. We show that if X is a Polish space, then the space B1(X,R) is normal iff X is countable.

Original language English 107248 Topology and its Applications 279 https://doi.org/10.1016/j.topol.2020.107248 Published - 1 Jul 2020

Keywords

• Baire function
• C(X,Y)
• Fréchet–Urysohn
• Function space
• Ideal of compact sets
• Metric space
• Normal
• Sequential
• cs-character
• k-space
• σ-space

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