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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?

Physical Mathematics
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    Yes, the question makes sense. The answer to it is "no". See this: https://math.stackexchange.com/questions/20913/are-continuous-self-bijections-of-connected-spaces-homeomorphisms – freakish Dec 27 '18 at 18:28
  • I can give you an example of a complex normed linear space $X$ and a continuous linear bijection $f:X\to X$ whose inverse is not continuous. – DanielWainfleet Jan 03 '19 at 10:26

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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.

Alex Ravsky
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