For a metric space $X$, let $\partial_\infty X$ denote the (Gromov) boundary at infinity of $X$. The following fact appears as an exercise in Bridson-Haefliger's standard text [1, p. 430].
Let $X$ be a geodesic space that is proper and hyperbolic. Then the natural map $\partial_\infty X \to Ends(X)$ is continuous and the fibres of this map are precisely the connected components of $\partial_\infty X$.
Thus, if I'm not mistaken, we have a natural 1-1 correspondence between the ends of $X$ and the connected components of $\partial_\infty X$.
This fact seems quite intuitive, so much so that the condition that $X$ be hyperbolic seems almost unnecessary. My question is, if we relax this condition, does this statement continue to be true, or even make sense anymore?
Edit: For convenience I add the relevant definitions below. Since we are working with spaces more general than hyperbolic, we have the following definition of the Gromov boundary. The following definition is taken from Section 8.3 of [2]
Definition. Let $X$ be a (quasi-)geodesic metric space. Then the Gromov boundary of $X$ is defined as $$ \partial X := \{\gamma : [0,\infty) \to X : \text{$\gamma$ is a quasi-geodesic ray}\} / \sim, $$ where we say that $\gamma \sim \gamma'$ if there exists some $c > 0$ such that (the image of) $\gamma$ is contained within the $c$-neighbourhood of $\gamma'$, and vice versa, i.e. they have finite Hausdorff distance.
We define a topology on $\partial X$ through convergence of sequences. We say that a sequence $(x_n)_n \in \partial X^{\mathbb N}$ converges to $x \in \partial X$ if there exists a sequence of quasi-geodesic rays $(\gamma_n)_n$ such that each $\gamma_n \in x_n$, and some $\gamma \in x$, such that every subsequence of $(\gamma_n)$ contains a subsequence that converges (uniformly on compact subsets of $[0,\infty)$) to $\gamma$.
[1] Bridson, M. R., & Haefliger, A. (2013). Metric spaces of non-positive curvature (Vol. 319). Springer Science & Business Media.
[2] Löh, C. (2017). Geometric group theory. Springer International Publishing AG.