I am self-studying topology from Munkres. One exercise asks, in part, to show that the spaces $(0,1)$ and $(0,1]$ are not homeomorphic. An apparent solution is as follows: If you remove a point, $x$, from $(0,1)$, you get a disconnected space; however, you can remove the point $\{1\}$ from the space $(0,1]$ and the space will still be connected. This apparently means the spaces aren't homeomorphic. I don't quite see the connection to the definition of homeomorphic spaces (that two spaces are homeomorphic if there is a bicontinuous function between them). So, what is the connection between the "bicontinuous" definition of homeomorphisms and the "removing a point" procedure? Sorry for missing something obvious.
General Topology – What Does Removing a Point Have to Do with Homeomorphisms?
general-topology
Best Answer
What you're missing is the following:
In particular, for any $x \in X$, $f|_{X\setminus\{x\}} : X\setminus\{x\} \to Y\setminus\{f(x)\}$ is a homeomorphism. Also note that the argument you mentioned uses the fact that if two spaces are homeomorphic, then they are either both connected or both disconnected.