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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 Fall '07
 Schilling
 Topology, Metric space, open balls, standard Euclidean metric

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