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][1]).
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.

  [1]: http://en.wikipedia.org/wiki/End_(topology)