## Abstract

The paper deals with Ascoli spaces C_{p}(X) and C_{k}(X) over Tychonoff spaces X. The class of Ascoli spaces X, i.e. spaces X for which any compact subset K of C_{k}(X) is evenly continuous, essentially includes the class of k_{R}-spaces. First we prove that if C_{p}(X) is Ascoli, then it is κ-Fréchet–Urysohn. If X is cosmic, then C_{p}(X) is Ascoli iff it is κ-Fréchet–Urysohn. This leads to the following extension of a result of Morishita: If for a Čech-complete space X the space C_{p}(X) is Ascoli, then X is scattered. If X is scattered and stratifiable, then C_{p}(X) is an Ascoli space. Consequently: (a) If X is a complete metrizable space, then C_{p}(X) is Ascoli iff X is scattered. (b) If X is a Čech-complete Lindelöf space, then C_{p}(X) is Ascoli iff X is scattered iff C_{p}(X) is Fréchet–Urysohn. Moreover, we prove that for a paracompact space X of point-countable type the following conditions are equivalent: (i) X is locally compact. (ii) C_{k}(X) is a k_{R}-space. (iii) C_{k}(X) is an Ascoli space. The Ascoli spaces C_{k}(X,I) are also studied.

Original language | English |
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Pages (from-to) | 35-50 |

Number of pages | 16 |

Journal | Topology and its Applications |

Volume | 214 |

DOIs | |

State | Published - 1 Dec 2016 |

## Keywords

- Ascoli
- C(X)
- C(X)
- Paracompact
- Scattered
- Stratifiable
- Čech-complete
- κ-Fréchet–Urysohn

## All Science Journal Classification (ASJC) codes

- Geometry and Topology