# hw8 - f X → Y Problem 4 Let f X → Y be a continuous map...

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Math 125A, Fall 2009 Homework 8 ONLY TURN IN PROBLEMS 1–4 Due Date: November 30, 2009 Problem 1: Let S be a subset of a metric space ( X,d ). (a) If S is the union of a collection of open balls, then prove that S is open. (b) If S is open, then prove that S can be written as a union of open balls. Problem 2: Determine whether or not the set is compact (see Deﬁnition 13.11). If the set is compact, then prove that it is. Otherwise, provide an open cover which doesn’t have a ﬁnite subcover. (a) (0 , 1) (b) [0 , 1] (c) { 5 } (d) R Problem 3: Let ( X,d X ) , ( Y,d Y ) be the metric spaces deﬁned by X = Y = R and: d X ( x,y ) = ( 1 , if x 6 = y 0 , if x = y and d Y ( x,y ) = | x - y | Note that ( Y,d Y ) is simply R equipped with the standard Euclidean metric. (a) Determine, with proof, the open sets of ( X,d X ). (b) Determine, with proof, all possible continuous functions
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Unformatted text preview: f : X → Y . Problem 4: Let f : X → Y be a continuous map between metric spaces ( X,d X ) and ( Y,d Y ). (a) Prove that f ( E ) ⊆ f ( E ) for any E ⊆ X . Recall that E represents the closure of E in X (see Deﬁnition 13.8 and Proposition 13.9). (b) By providing an example, show that f ( E ) can be a proper subset of f ( E ) . Problem 5: Prove that { ( x,y ) : x 2 + y 2 = 1 } is a connected subset of R 2 . Problem 6: Let E,F be subsets of some metric space ( X,d ) and suppose E ∩ F 6 = ∅ . (a) If E,F are connected, then prove that E ∪ F is connected. (b) If E,F are path connected, then prove that E ∪ F is path connected....
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## This note was uploaded on 03/18/2012 for the course MATH 67 taught by Professor Schilling during the Fall '07 term at UC Davis.

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