245B_W09_hwk5

245B_W09_hwk5 - 245B Winter 2009 Assignment 5 Notes and selected model answers(Model solutions follow question numbers in bold Folland Chapter 4 28

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Unformatted text preview: 245B, Winter 2009, Assignment 5: Notes and selected model answers (Model solutions follow question numbers in bold .) Folland Chapter 4 28. a. We must verify that T satisfies the axioms of a topology. 1. We always have π- 1 ( ∅ ) = ∅ and π- 1 ( ˜ X ) = X , and both of these pre- images are open in X , so ∅ , ˜ X ∈ T ; 2. If U is any collection of members of T , then by definition { π- 1 ( U ) : U ∈ U} is a collection of open subsets of X . Therefore S U ∈U π- 1 ( U ) is a union of open sets in X and so is itself open, but since π- 1 [ U = [ U ∈U π- 1 ( U ) this tells us that π- 1 (S U ) is open, and so S U ∈ T ; 3. Exactly similarly, if U is a finite family of members of T then π- 1 U = U ∈U π- 1 ( U ) is now a finite intersection of open subsets of X , so is itself open, and so the finite intersection T U also lies in T . b. Suppose that Y is another topological space and f : ˜ X → Y a function. First note that if U ⊆ ˜ X is open (i.e., it lies in the family T that we have shown above to be a topology) then by the definition of...
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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.

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245B_W09_hwk5 - 245B Winter 2009 Assignment 5 Notes and selected model answers(Model solutions follow question numbers in bold Folland Chapter 4 28

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