What is the one point compactification of $X=(-1,0)\cup (0,1)$?
I think that the answer might be two circles in the plane sharing one tangent point but I don't know how to prove it properly.
What I thought was start with the set $(-1,0]\cup [0,1)$ obtained by adjoining the point $0$ to $X$. That (I thought) is homeomorphic to two circles with a common tangen point, since if we take that tangent point we should get two disjoint open intervals on the real line which I thought it might be homeomorphic to my original space $X$ because any two open intervals are homeomorphic.
If this is right, how can I translate it into proper mathematical language? Thanks in advance!
The one-point compactification of $(-1,0)\cup(0,1)$ is indeed the wedge of two circles. You can prove it as follows:
As a set, the compactification is $(-1,0)\cup(0,1)\cup\{*\}$. Map it to $S^1\vee S^1$ by sending $*$ to the point where the two circles join and the two segments homeomorphically onto the two circles missing a point. This map is obviously bijective. Now show that both it and its inverse are continuous using what you know about the topology of the compactification.