Let $X$ be a proper geodesic metric space that is $\delta$-hyperbolic.

Definition. We define the Gromov boundary $\partial X$ of $X$ as the set of all the geodesic rays $c:[0, \infty)\to X$, where we regard two geodesic rays $c$ and $c'$ as the same if the Hausdorff distance between then is finite. We denote the equivalence class of a geodesic ray $c$ by $c(\infty)$.

Now we have a set $\bar X:=X\cup \partial X$. We want to define a topology on it. Before that we need to make a definition.

Definition. A generalized ray in $X$ is a geodesic $c:[0, T]\to X$, which we extend on $[0, \infty]$ by definition $c(t)=c(T)$ for all $t>T$ (So $c(\infty)=c(T)$ also).

Definition. We define a topology on $\bar X$ as follows. Fix a point $p\in X$. For a sequence of points $(x_n)$ in $\bar X$, and a point $x$ in $\bar X$, we write $x_n\to x$ if there is a sequence of generalized rays $(c_n)$ in $X$, with $c_n(0)=p$ and $c_n(\infty)=x_n$, such that every subsequence of $(c_n)$ has a subsequence which converges uniformly on compact sets to a generalized ray $c$ satisfying $c(\infty)=x$.

This defines a topology as follows: The closed sets are precisely those subsets $C$ of $\bar X$ for which the following holds: Whenever $(x_n)$ is a sequence in $C$ with $x_n\to x$ for a point $x\in \bar X$, then $x\in C$.

Now here is my question.

Question. Why this topology is independent of the choice of the point $p$?


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