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

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