Unformatted text preview: x ∈ X and any ε > , there is a δ > such that, setting y = f ( x ) , N δ ( x ) ⊂ f1 ( N ε ( y )) . 4. Prove the following. Theorem. Let ( X,d x ) and ( Y,d Y ) be metric spaces. Let f : X → Y be continuous. For any compact set A ⊂ X , f ( A ) is compact. 5. Let { x t } be a sequence in R and let E denote the set of subsequential limits of { x t } . (a) Provide an example in which E = [0 , 1]. (b) Provide an example in which E = R . (c) Prove that it is impossible to have E = (0 , 1)....
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 Fall '08
 JohnNachbar
 Economics, Topology, Metric space, Compact space, Closed set

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