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Theorem 26.6. Let f : X ? Y be a bijective continuous function. If X is compact and Y is Hausdorff, then f is a homeomorphism. Proof. We shall prove that images of closed sets of X under f are closed in Y; this will prove continuity of the map f^{-1}. If A is closed in X, then A is compact, by Theorem 26.2. Therefore, by the theorem just proved, f(A) is compact. Since Y is Hausdorff, f(A) is closed in Y, by Theorem 26.3.

Name: Theorem 26.6. Let f : X ? Y be a bijective continuous function. If X is compact and Y is Hausdorff, then f is a homeomorphism. Proof. We shall prove that images of closed sets of X under f are closed in Y; this will prove continuity of the map f^-1. If A is closed in X, then A is compact, by Theorem 26.2. Therefore, by the theorem just proved, f(A) is compact. Since Y is Hausdorff, f(A) is closed in Y, by Theorem 26.3.
Uploaded: 2022-11-24T17:10:50-08:00
Duration: 1 min 35 s
Channel: Sri K

          Theorem 26.6. Let f : X ? Y be a bijective continuous function. If X is compact and Y is Hausdorff, then f is a homeomorphism. Proof. We shall prove that images of closed sets of X under f are closed in Y; this will prove continuity of the map f^{-1}. If A is closed in X, then A is compact, by Theorem 26.2. Therefore, by the theorem just proved, f(A) is compact. Since Y is Hausdorff, f(A) is closed in Y, by Theorem 26.3.

Theorem 26.6. Let f : X ? Y be a bijective continuous function. If X is compact and Y is Hausdorff, then f is a homeomorphism. Proof. We shall prove that images of closed sets of X under f are closed in Y; this will prove continuity of the map f^-1. If A is closed in X, then A is compact, by Theorem 26.2. Therefore, by the theorem just proved, f(A) is compact. Since Y is Hausdorff, f(A) is closed in Y, by Theorem 26.3.

Added by James H.

Calculus: Early Transcendentals

James Stewart 8th Edition

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Solved by Expert Sri K

11/24/2022

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We are given a bijective continuous function \( f: X \to Y \), where \( X \) is compact and \( Y \) is Hausdorff. We need to prove that \( f \) is a homeomorphism, which means we need to show that \( f \) is a continuous bijection with a continuous inverse. Show more…

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Theorem 26.6. Let f : X → Y be a bijective continuous function. If X is compact and Y is Hausdorff, then f is a homeomorphism. Proof. We shall prove that images of closed sets of X under f are closed in Y; this will prove continuity of the map f^-1. If A is closed in X, then A is compact, by Theorem 26.2. Therefore, by the theorem just proved, f(A) is compact. Since Y is Hausdorff, f(A) is closed in Y, by Theorem 26.3.