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

For a separable locally compact but not compact metrizable space $X$, let $\alpha X=X \cup\left\{x_{\infty}\right\}$ be the one-point compactification with the point at infinity $x_{\infty}$. We denote by $\operatorname{EM}(X)$ 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, and there exists a sequence $\left(C_i\right)_{i \in \mathbb{N}}$ 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 $i \geq i(K), C_i \subset X \backslash K$, then $\operatorname{EM}(X)$ is homeomorphic to $\mathbf{c}_0$.