Theorem 7.29. Suppose f : X → Y is a continuous bijection where X is compact and Y is Hausdorff. Then f is a homeomorphism.

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Definition. A function f : X → Y is a homeomorphism if and only if f is a continu-
ous bijection and the inverse map f-1 : Y → X is also continuous.
Definition. Two topological spaces, X and Y, are homeomorphic or topologically
equivalent if and only if there exists a homeomorphism f : X → Y.
Theorem 7.29. Suppose f : X → Y is a continuous bijection where X is compact and
Y is Hausdorff. Then f is a homeomorphism.
Transcribed Image Text:Definition. A function f : X → Y is a homeomorphism if and only if f is a continu- ous bijection and the inverse map f-1 : Y → X is also continuous. Definition. Two topological spaces, X and Y, are homeomorphic or topologically equivalent if and only if there exists a homeomorphism f : X → Y. Theorem 7.29. Suppose f : X → Y is a continuous bijection where X is compact and Y is Hausdorff. Then f is a homeomorphism.
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