204PS32008 - Econ 204 Summer/Fall 2008 Problem Set 3 Due in Lecture Friday August 8 1 Show that if f A → R n is continuous and B ⊂ A then the

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Unformatted text preview: Econ 204, Summer/Fall 2008 Problem Set 3 Due in Lecture Friday, August 8 1. Show that if f : A → R n is continuous, and B ⊂ A , then the restriction f | B is continuous. 1 2. Let { S i } i ∈ I be a collection of connected subsets of a space X . Suppose there exists an i ∈ I such that for each i ∈ I , the sets S i and S i have non-empty intersection. Show that S i ∈ I S i is connected. 3. Determine whether the following subsets of R are compact or not and justify your statements: (a) N (b) { } ∪ { 1 n : n ∈ N } 4. Prove the following statements related to compactness: (a) Show that the union of finitely many compact sets is compact. Equivalently, if every set A k from { A k : 1 ≤ k ≤ N } is compact, then ∪ 1 ≤ k ≤ N A k is compact. (b) Show that the intersection of any family of compact sets is compact. Equivalently, if every set A λ from { A λ : λ ∈ Λ } is compact, then ∩ λ ∈ Λ A λ is compact....
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This note was uploaded on 08/01/2008 for the course ECON 204 taught by Professor Anderson during the Fall '08 term at University of California, Berkeley.

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204PS32008 - Econ 204 Summer/Fall 2008 Problem Set 3 Due in Lecture Friday August 8 1 Show that if f A → R n is continuous and B ⊂ A then the

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