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Unformatted text preview: MATH 202A HOMEWORK 8 The following assignment is due Wed., Oct. 24. Recall we define a space X to be locally compact if for each x ∈ X , there is a neighborhood U of x such that ¯ U is compact. Problem 1 . (a) Let X α be a Hausdorff topological space for each α ∈ I . Prove that if an infinite number of the coordinate spaces X α are non-compact, then each compact subset of the product Q α ∈ I X α is nowhere dense. [We conclude that Q α X α is not locally compact.] Hint: The projection map onto a coordinate is continuous, and thus maps compact sets to compact sets. (b) A continuous map need not map a locally compact space to a locally compact space. Hint: Any space with the discrete topology is locally compact. (c) Suppose f : X → Y is continuous, open, and surjective, X is locally compact and Y is Hausdorff. Prove that Y is locally compact. [This is summarized by saying that an open continuous function maps a locally compact space to a locally compact space]....
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This note was uploaded on 04/06/2010 for the course MATH various taught by Professor Tao/analysis during the Spring '10 term at UCLA.
- Spring '10