Possible Duplicate:
Are continuous self-bijections of connected spaces homeomorphisms?
I know that a function $f\colon X\to Y$ need not be a homeomorphism if it is:
- A continuous bijection (e.g. $f:[0,2\pi)\to S^1$),
- An open bijection (e.g. inverse of above function),
- A bijection between homeomorphic spaces (e.g. $x\mapsto -x$ on $\mathbb R$ with lower limit topology),
but what if $X,Y$ are already known to be homeomorphic, and $f$ is a continuous bijection. Must f then be a homeomorphism, or does there exist a counterexample?
I have given this some thought, but I have so far been unsuccessful at generating either a proof or a counterexample. Also, my searches have turned up unsuccessful.