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 languageEnglish
Article number107248
JournalTopology and its Applications
Volume279
DOIs
StatePublished - 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

Fingerprint

Dive into the research topics of 'Topological properties of some function spaces'. Together they form a unique fingerprint.

Cite this