Question

Problem 5.7. Let f : K ? Y be a continuous map from a compact space K to a Hausdorff space Y. Prove that if f is one-to-one and onto, then f is a homeomorphism. Hint: The hypothesis implies that the inverse g = f?! exists. First show that for any ? ? K g?!(?) = f(?). Complete the proof using the facts below (you do not have to prove these). Fact A. A closed subset of a compact space is itself compact. Fact B. A compact subset of a Hausdorff space is closed. Fact C. The image of a compact set under a continuous map is compact (Lemma 38). Note: In class, we proved Facts A and B for metric spaces (Prop. 17 and Cor. 18). The proofs for topological spaces is essentially the same (Gamelin & Greene pages 79-80).

          Problem 5.7. Let f : K ? Y be a continuous map from a compact space K to a Hausdorff space Y. Prove that if f is one-to-one and onto, then f is a homeomorphism.

Hint: The hypothesis implies that the inverse g = f?! exists. First show that for any ? ? K
g?!(?) = f(?).

Complete the proof using the facts below (you do not have to prove these).

Fact A. A closed subset of a compact space is itself compact.
Fact B. A compact subset of a Hausdorff space is closed.
Fact C. The image of a compact set under a continuous map is compact (Lemma 38).

Note: In class, we proved Facts A and B for metric spaces (Prop. 17 and Cor. 18). The proofs for topological spaces is essentially the same (Gamelin & Greene pages 79-80).

Problem 5.7. Let f : K ? Y be a continuous map from a compact space K to a Hausdorff space Y. Prove that if f is one-to-one and onto, then f is a homeomorphism.

Hint: The hypothesis implies that the inverse g = f?! exists. First show that for any ? ? K
g?!(?) = f(?).

Complete the proof using the facts below (you do not have to prove these).

Fact A. A closed subset of a compact space is itself compact.
Fact B. A compact subset of a Hausdorff space is closed.
Fact C. The image of a compact set under a continuous map is compact (Lemma 38).

Note: In class, we proved Facts A and B for metric spaces (Prop. 17 and Cor. 18). The proofs for topological spaces is essentially the same (Gamelin Greene pages 79-80).

Added by David A.

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