# Does a continuous bijection from a compact, hausdorff space imply it is an homeomorphism?

Let $$f:X \rightarrow Y$$ be a bijective, continuous map between two toplogical spaces. Does X being compact and Hausdorff imply that $$f$$ must be a homeomorphism? I think it doesn't but I can not find an example.

Let $$X$$ be your favourite compact Hausdorff space with at least $$2$$ points, let $$Y$$ be $$X$$ with the trivial topology, consider the identity as $$f$$.
• I think you should specify what you mean by trivial: is it the discrete topology or the anti-discrete topology? This clearly cannot be true for the discrete topology since $f$ itself need not be continuous. – weierstrash Jun 26 at 16:27