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Title: | Topological properties of some function spaces |

Authors: | Gabriyelyan, S. Osipov, A. V. |

Issue Date: | 2020 |

Publisher: | Elsevier B.V. |

Citation: | Gabriyelyan S. Topological properties of some function spaces / S. Gabriyelyan, A. V. Osipov. — DOI 10.1016/j.topol.2020.107248 // Topology and its Applications. — 2020. — Vol. 279. — 107248. |

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 Xℵ0 has the property γ, where Xℵ0 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. © 2020 Elsevier B.V. |

Keywords: | BAIRE FUNCTION CP(X,Y) CS⁎-CHARACTER FRÉCHET–URYSOHN FUNCTION SPACE IDEAL OF COMPACT SETS K-SPACE METRIC SPACE NORMAL SEQUENTIAL Σ-SPACE |

URI: | http://hdl.handle.net/10995/101529 |

Access: | info:eu-repo/semantics/openAccess |

SCOPUS ID: | 85084351854 |

PURE ID: | 12905129 632ed029-7655-4d45-82f0-c17a1e5a621a |

ISSN: | 1668641 |

DOI: | 10.1016/j.topol.2020.107248 |

Appears in Collections: | Научные публикации, проиндексированные в SCOPUS и WoS CC |

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