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Let f: X → Y be a continuous map from a compact space X to a Hausdorff space Y. Let C be a closed subspace of Y, and let U be an open neighborhood of f-¹(C) in X. Show that there is an open neighborhood V of C in Y such that f-¹(V) is contained in U.

Let f: X → Y be a continuous map from a compact space X to a Hausdorff space Y. Let C be a closed subspace of Y, and let U be an open neighborhood of f-¹(C) in X. Show that there is an open neighborhood V of C in Y such that f-¹(V) is contained in U.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Definition 1. Let X and Y be two topological spaces. A map f: X→ Y is proper if the preimage under f of any compact set is compact. Definition 2. A topological space X is said to be compactly generated if the following condition is satisfied: A subspace A is closed in X if and only if An K is closed in K for all compact subspaces KC X. Definition 3. A topological space X is locally compact if the following condition is satis- fied: For every point x E X, there is a compact subset KCX that contains an (open) neighborhood of x. Let f: X → Y be a continuous map from a compact space X to a Hausdorff space Y. Let C be a closed subspace of Y, and let U be an open neighborhood of f-¹(C) in X. Show that there is an open neighborhood V of C in Y such that f-¹(V) is contained in U.
Transcribed Image Text:Definition 1. Let X and Y be two topological spaces. A map f: X→ Y is proper if the preimage under f of any compact set is compact. Definition 2. A topological space X is said to be compactly generated if the following condition is satisfied: A subspace A is closed in X if and only if An K is closed in K for all compact subspaces KC X. Definition 3. A topological space X is locally compact if the following condition is satis- fied: For every point x E X, there is a compact subset KCX that contains an (open) neighborhood of x. Let f: X → Y be a continuous map from a compact space X to a Hausdorff space Y. Let C be a closed subspace of Y, and let U be an open neighborhood of f-¹(C) in X. Show that there is an open neighborhood V of C in Y such that f-¹(V) is contained in U.
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