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

TL;DR

This paper characterizes the topological structure of the space of admissible metrics on certain non-compact, locally connected, separable metrizable spaces, showing it is homeomorphic to the space of sequences converging to zero.

Contribution

It establishes conditions under which the space of admissible metrics on such spaces is topologically equivalent to c0, extending understanding of metric spaces with the compact-open topology.

Findings

EM(X) is homeomorphic to c0 under specified conditions.

The space of admissible metrics has the topology of sequences converging to zero.

Provides a topological classification for metric spaces with certain connectedness properties.

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 $|i - j| \leq 1$, $C_i \cap C_j \neq \emptyset$, and for each compact set $K \subset X$, there is a positive integer $i(K) \in \mathbb{N}$ such that for any $i \geq i(K)$, $C_i \subset X \setminus K$, then $EM(X)$ is homeomorphic to $ \mathbf{c}_0$.