Topological properties of function spaces $C_k(X,2)$ over zero-dimensional metric spaces $X$

TL;DR

This paper characterizes the topological properties of the function space $C_k(X,2)$ over zero-dimensional metric spaces $X$, linking these properties to the local compactness and separability of $X$ and its derived set.

Contribution

It provides a comprehensive classification of various topological properties of $C_k(X,2)$ based on the structure of $X$, including conditions for being Ascoli, $k$-space, sequential, Fréchet-Urysohn, normal, and having countable tightness.

Findings

$C_k(X,2)$ is Ascoli iff $X$ is locally compact or $X'$ is compact.

$C_k(X,2)$ is a $k$-space iff $X$ is a sum of a Polish locally compact space and a discrete space or $X'$ is compact.

$C_k(X,2)$ is sequential iff $X$ is Polish and locally compact or $X'$ is compact.

Abstract

Let $X$ be a zero-dimensional metric space and $X'$ its derived set. We prove the following assertions: (1) the space $C_k(X,2)$ is an Ascoli space iff $C_k(X,2)$ is $k_\mathbb{R}$-space iff either $X$ is locally compact or $X$ is not locally compact but $X'$ is compact, (2) $C_k(X,2)$ is a $k$-space iff either $X$ is a topological sum of a Polish locally compact space and a discrete space or $X$ is not locally compact but $X'$ is compact, (3) $C_k(X,2)$ is a sequential space iff $X$ is a Polish space and either $X$ is locally compact or $X$ is not locally compact but $X'$ is compact, (4) $C_k(X,2)$ is a Fr\'{e}chet--Urysohn space iff $C_k(X,2)$ is a Polish space iff $X$ is a Polish locally compact space, (5) $C_k(X,2)$ is normal iff $X'$ is separable, (6) $C_k(X,2)$ has countable tightness iff $X$ is separable. In cases (1)-(3) we obtain also a topological and algebraical structure of $C_k(X,2)$.