In this post I give examples of topological spaces for which bijectively relations imply existence of an homeomorphism. Namely:
I also give a counterexample of bijectively related spaces for which an homeomorphism doesn't exist.
What are additional properties that topological spaces can have and that force an homeomorphism to exist if those spaces are bijectively related?
One can generalize the notion of compactness so that the property that bijection implies homeomorphism still holds. Suppose that $\kappa$ is a regular cardinal. Then a completely regular space $X$ is said to be a $P_{\kappa}$-space if whenever $|I|<\kappa$ and $U_{i}$ is open for each $i\in I$, then $\bigcap_{i\in I}U_{i}$ is open as well. A topological space $X$ is said to be $\kappa$-compact if every open cover $\mathcal{U}$ of $X$ has a subcover $\mathcal{V}$ with $|\mathcal{V}|<\kappa$. Then whenever $X,Y$ are $\kappa$-compact $P_{\kappa}$-space and $f:X\rightarrow Y$ are continuous bijections, then $f$ is a homeomorphism. The proof is exactly the same as the proof using compact Hausdorff spaces. In fact, the much of the basic theory of $\kappa$-compact $P_{\kappa}$-spaces is identical the the basic theory of compact Hausdorff spaces, so many results about compact Hausdorff spaces also apply to $\kappa$-compact $P_{\kappa}$-spaces.
The class $\ \mathbf M_h\ $ of minimal Hausdorff spaces is properly larger than the class of the Hausdorff compact spaces. Every continuous bijection of a Hausdorff minimal space onto arbitrary Hausdorff space is a homeomorphism. This is really a simple tautology. The same holds for other minimal spaces within other classes.
It is a somewhat trivial observation that if a space $X$ is reversible (i.e. every continuous bijection $X\to X$ is a homeomorphism) then every space bijectively related to $X$ is homeomorphic to $X$.
There is some literature on this subject (please forgive me the self-promotion):
There is also a conference paper "Unusual and bijectively related manifolds" by J. G. Hocking, but I cannot find it.
On MathOverflow, apart from the question linked to by The Masked Avenger, see also Continuous bijections vs. Homeomorphisms.
