The topological type of spaces consisting of certain metrics on locally compact metrizable spaces with the compact-open topology

Abstract

For a separable locally compact but not compact metrizable space $X$, let $\alpha X=X\cup \{x_{\infty }\}$ be the one-point compactification with the point at infinity $x_{\infty }$. We denote by $EM(X)$ the space consisting of admissible metrics on $X$, which can be extended to an admissible metric on $\alpha X$, endowed with the compact-open topology. Let ${\mathbf{c}}_0\subset (0,1{)}^{\mathbb{N}}$ be the space of sequences converging to $0$. In this paper, we shall show that if $X$ is separable, locally connected and locally compact but not compact, and there exists a sequence $\{C_i\}$ of connected sets in $X$ such that for all positive integers $i,j\in \mathbb{N}$ with $|i-j|\le 1$, $C_i\cap C_j\ne \varnothing$, and for each compact set $K\subset X$, there is a positive integer $i(K)\in \mathbb{N}$ such that for any $i\ge i(K)$, $C_i\subset X\setminus K$, then $EM(X)$ is homeomorphic to ${\mathbf{c}}_0$.