TITLE: Main Theorems/Techniques for proving Homeomorphism?
QUESTION [13 upvotes]: General Question: what are the most common Theorems/Methods used to prove Homeomorphism?
I encountered: 

- find the map explicitly
- use the Compact-to-Hausdorff Lemma
- find cts maps $f$ and $g$ s.t. $f\circ g=g\circ f= i$ where $i$ is the identity map.

Can anyone deepen/correct/enlarge my list?

REPLY [7 votes]: Suppose that $f: X \rightarrow Y$ is a bijective continuous map. In general, from this it does not follow that $f^{-1}$ is continuous, so $f$ is not necessarily a homeomorphism.
However, there are some special cases where $f: X \rightarrow Y$ bijective and continuous implies that $f$ is a homeomorphism. In any introductory course on functional analysis, one comes across to the open mapping theorem. As an easy corollary, you get the bounded inverse theorem:

Bounded Inverse Theorem: Let $X$ and $Y$ be Banach spaces. Suppose that $T: X \rightarrow Y$ is a bijective, linear and continuous map. Then $T$ is a homeomorphism.

This is a very special case, here linearity and the algebraic structure of $X$ and $Y$ is vital for the above theorem to hold. There is also the following theorem, proven in the 1960's.

Theorem: Any two separable infinite-dimensional Banach spaces are homeomorphic.