The Ascoli property for function spaces

TL;DR

This paper investigates the Ascoli property in function spaces over Tychonoff spaces, establishing conditions under which these spaces are Ascoli, Fréchet-Urysohn, or scattered, and extending classical results in topology.

Contribution

It characterizes when function spaces $C_p(X)$ and $C_k(X)$ are Ascoli, linking this property to other topological features like being $ ext{k}_{ ext{R}}$-spaces, scattered, or Fréchet-Urysohn, and extends known results for specific classes of spaces.

Findings

$C_p(X)$ is Ascoli iff it is $ ext{k}_{ ext{F}}$-space for cosmic $X$.

$C_p(X)$ is Ascoli iff $X$ is scattered for certain classes of spaces.

Equivalence of local compactness, $k_{ ext{R}}$-space, and Ascoli property for $C_k(X)$ on paracompact spaces of point-countable type.

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 $k_{\mathbb R}$-spaces. First we prove that if $C_p(X)$ is Ascoli, then it is $\kappa$-Fr\'echet-Urysohn. If $X$ is cosmic, then $C_p(X)$ is Ascoli iff it is $\kappa$-Fr'echet-Urysohn. This leads to the following extension of a result of Morishita: If for a \v{C}ech-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 \v{C}ech-complete Lindel\"of space, then $C_p(X)$ is Ascoli iff $X$ is scattered iff $C_p(X)$ is Fr\'echet-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 $k_{\mathbb R}$-space. (iii) $C_k(X)$ is an Ascoli space. The Asoli spaces $C_k(X,[0,1])$ are also studied.