Continuous bijections and their inverses

Question

Are there any theorems characterizing the continuity of the inverses of bijections between certain classes of particular spaces. E.g. it seems to me the inverse of any continuous bijection between connected subsets of $\mathbb{R}$ is continouos. It seems that most counterexamples require highly specific spaces. I'm thinking there might be something along the lines of any continuous bijection between homeomorphic spaces has a continuous inverse. Any of this make sense?

Answer 1

Indeed, in particular cases the inverse of a continuous bijection $f:X\to Y$ is continuous. Namely, when $X$ is a compact, $Y$ is Hausdorff (well-known and very easy to prove) or when $X$ is an open subset of $\Bbb R^n$ and $Y\subset\Bbb R^n$, by invariance of domain.