Discussion
Let X and Y be topological spaces and \(f : X \to Y\) be a continuous and bijective mapping. Which one of the following statements is TRUE?
Answer: C
The true statement is f is a homeomorphism if X is compact and Y is Hausdorff.
It suffices to show f is a closed map. A closed subset of the compact space X is compact; its continuous image is compact; and a compact subset of the Hausdorff space Y is closed. So f maps closed sets to closed sets, which makes \(f^{-1}\) continuous.
The other conditions are not enough. The identity from [0,1] with the usual topology onto [0,1] with the indiscrete topology is a continuous bijection between compact spaces that is not a homeomorphism; the same map shows that a Hausdorff domain with a compact codomain fails. The map \(t \mapsto (\cos t, \sin t)\) from \([0, 2\pi)\) onto the circle is a continuous bijection between Hausdorff spaces whose inverse is not continuous.
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