## The Ascoli property for function spaces

### Authors:

- Saak Gabriyelyan,
- Jan Grebík,
- Jerzy Kąkol,
- Lyubomyr Zdomskyy

### 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.

- Record ID
- UAM689de566d0b74b169189c3a5b58645a2
- Author
- Journal series
- Topology and Its Applications, ISSN 0166-8641
- Issue year
- 2016
- Vol
- 214
- Pages
- 35-50
- ASJC Classification
- DOI
- DOI:10.1016/j.topol.2016.08.026 opening in a new tab
- Language
- en English
- Score (nominal)
- 20
- Score source
- journalList
- Score
- Publication indicators
- = 4; : 2016 = 0.913; : 2016 = 0.377 (2) - 2016=0.464 (5)

- Uniform Resource Identifier
- https://researchportal.amu.edu.pl/info/article/UAM689de566d0b74b169189c3a5b58645a2/

* presented citation count is obtained through Internet information analysis and it is close to the number calculated by the Publish or Perishopening in a new tab system.