For non-arbitary spaces we can discuss for such case, like how many components are there or other properties. But is it true for any space? It seems if we have a homeomorphism $f$ from $S$ to $S' = S – \{p\}$, $f(p) = q$, but since a space is homeomorphic to itself, there is some $g(r)=q$. Then there is no inverse if $f$ and $g$ coincide. However they don't have to and maybe $f$ is somehow the homeomorphism since I can't deduce more information.
If it is not true, a counter-example will be super helpful! Thank you.
Best Answer
Another example: $\mathbb C \setminus \mathbb Z$. This is connected and homeomorphic to the punctured version of itself.