Freundenthal introduced ends of topological spaces and the end point compactification of locally compact topological spaces adding one point for each end of the topological space (see here). For example the end point compactification of $\mathbb{R}$ is homeomorphic to the unit interval $[0,1]$ because "$\mathbb{R}$ has two ends".

If you take the set $A=\{ (x, 1/x)\ |\ x>0 \}\cup\{(x, 0)\ |\ x\in\mathbb{R}\}$, it is homeomorphic to $\mathbb{R}\sqcup\mathbb{R}$ so its end point compactification will be homeomorphic to $[0,1]\sqcup[0,1]$.

But seeing $A$ as a metric space one wants to say that "the two ends at the right are the same" and to compactify $A$ with only three ends, giving a connected topological space homeomorphic to $[0,1]$.

Is there in the litterature such a notion of "metric end point compactification" which would compactify $A$ with only three ends?

(I’m not asking how to define such a compactification, I already have a definition which seems to work for my purpose, I just want to know if something like that is already known)

Thank you

Edit: Here is a more precise definition of the compactification I need (I’m not sure this is exactly the correct definition, but it should be something like that): let’s say that a metric space $(E,d)$ is 0-connected if for all $\varepsilon>0$ and $x,y\in E$, there exists a finite sequence $(u_0, \dots, u_n)$ such that $u_0 = x$, $u_n = y$ and $d(u_i,u_{i+1})<\varepsilon$ for all $0\leqslant i\leqslant n-1$ (every connected space is 0-connected, but the space $A$ introduced before is an example of space which is non connected but 0-connected) (by the way, if this notion of 0-connectedness has already a name in the litterature, I will be happy to know it).

Then, take the definiton of the Freudenthal end point compactification (see link above) but replace "$U_i$ is a connected component of …" by "$U_i$ is a 0-connected component of …".

In particular $A$ has three ends, and the complex plane or the hyperbolic space have only one end.