Continuous Bijection From Compact Space to Hausdorff Space is Homeomorphism
If \(f : X \to Y\) is a continuous bijection between a compact space to a Hasdorff space, then \(f\) is a homeomorphism.
The idea here is that with additional restrictions on the properties of the domain and codomain, we don't need to check that the inverse is bijective.
Proof
By the definition of a homeomorphism, we just need to prove that \(g = f^{-1}\) is continuous (we introduce \(g\) to prevent confusion with pre-images and inverse functions). We do this with the closed set definition of continuity, that is \(g^{-1}(S)\) is closed for all closed \(S\). This is equivalent to showing that \(f(S)\) is closed for all \(S\), that is, that \(f\) is a closed mapping, because \(f\) is bijection.
Therefore, let \(S\) be a closed subset of \(X\). \(S\) is therefore compact given that \(X\) is compact. Hence because compactness is preserved under continuous functions, we know that \(f(S)\) must also be compact, and therefore every compact subset of a hausdorff space is closed|as a subset of a Hausdorff space which is compact, \(f(S)\) is closed.