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.

bijectively related? $\endgroup$ – Włodzimierz Holsztyński Apr 9 '15 at 8:34