So we all know the basic fact that for two topological spaces $X$ and $Y$ such that $X$ is compact and $Y$ is hausdorff then every bijective continuous map $ f: X \rightarrow Y$ is a homeomorphism.
What I was wondering is: Under what kind on general conditions on $X$ and $Y$ does the following holds ?
$$ \forall f\in C(X,Y) \text{ a bijection } : f^{-1} \in C(Y,X)$$
I've tried to find conditions on my own but it turns out that it's pretty hard to have good intuition about which conditions will work. For example it feels like the statement should be true for $f: \mathbb{R}^n \rightarrow \mathbb{R}^n$ continuous and bijective but I haven't been able to prove this.
