TITLE: Whether $(0,1)$ and $(0,1]$ are homeomorphic
QUESTION [8 upvotes]: In connection with the question continuous onto map from $(0,1)\to (0,1]$ I would like to know whether $(0,1)$ and $(0,1]$ are homeomorphic. The map mentioned in the above question is onto but not a bijection. So does such a continuous bijection exist?

REPLY [13 votes]: HINT: For every $x\in(0,1)$, $(0,1)\setminus\{x\}$ is not connected. If $(0,1)$ and $(0,1]$ were homeomorphic, the same would be true of $(0,1]$, since connectedness is a topological property. Does $(0,1]$ actually have this property, or does it have some point that can be removed without disconnecting it?