## Abstract

Let Y be a metrizable space containing at least two points, and let X be a Y_{I}-Tychonoff space for some ideal I of compact sets of X. Denote by C_{I}(X,Y) the space of continuous functions from X to Y endowed with the I-open topology. We prove that C_{I}(X,Y) is Fréchet–Urysohn iff X has the property γ_{I}. We characterize zero-dimensional Tychonoff spaces X for which the space C_{I}(X,2) is sequential. Extending the classical theorems of Gerlits, Nagy and Pytkeev we show that if Y is not compact, then C_{p}(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 B_{1}(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 B_{1}(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 X_{ℵ0 } has the property γ, where X_{ℵ0 } is the space X with the Baire topology. We show that if X is a Polish space, then the space B_{1}(X,R) is normal iff X is countable.

Original language | English |
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Article number | 107248 |

Journal | Topology and its Applications |

Volume | 279 |

DOIs | |

State | 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