## Abstract

A Tychonoff space X is called (sequentially) Ascoli if every compact subset (resp. convergent sequence) of C_{k}(X) is evenly continuous, where C_{k}(X) denotes the space of all real-valued continuous functions on X endowed with the compact-open topology. Various properties of (sequentially) Ascoli spaces are studied, and we give several characterizations of sequentially Ascoli spaces. Strengthening a result of Arhangel'skii we show that a hereditary Ascoli space is Fréchet–Urysohn. A locally compact abelian group G with the Bohr topology is sequentially Ascoli iff G is compact. If X is totally countably compact or near sequentially compact then it is a sequentially Ascoli space. The product of a locally compact space and an Ascoli space is Ascoli. If additionally X is a μ-space, then X is locally compact iff the product of X with any Ascoli space is an Ascoli space. Extending one of the main results of [18] and [16] we show that C_{p}(X) is sequentially Ascoli iff X has the property (κ). We give a necessary condition on X for which the space C_{k}(X) is sequentially Ascoli. For every metrizable abelian group Y, Y-Tychonoff space X, and nonzero countable ordinal α, the space B_{α}(X,Y) of Baire-α functions from X to Y is κ-Fréchet–Urysohn and hence Ascoli.

Original language | English |
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Article number | 107401 |

Journal | Topology and its Applications |

Volume | 285 |

DOIs | |

State | Published - 1 Nov 2020 |

## Keywords

- Ascoli
- Baire-α function
- C(X)
- C(X)
- P-space
- Sequentially Ascoli
- κ-Fréchet–Urysohn

## ASJC Scopus subject areas

- Geometry and Topology