204ps32006 - Economics 204 August 2006 Due Friday 11 August...

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Problem Set 3 Economics 204 - August 2006 Due Friday, 11 August in Lecture 1. Connectedness (a) Suppose we define two sets as being separated if neither contains a limit point of the other and that a set in a metric space is connected if it cannot be written as the union of two disjoint non-empty separated sets. Prove that this definition is equivalent to the one given in lecture. (b) Suppose that the sets C and D are two non-empty separated subsets of X whose union is X and Y is a connected subset of X . Prove that Y lies entirely within C or D . (c) Prove that the union of a collection of connected sets that have a point in common is connected. 2. For each set, exhibit an open cover with no finite subcover. (a) (0 , 1) (b) R 3. Use the open cover definition of compactness to prove that every sequence in a compact set has a convergent subsequence. 4. Use the open cover definition of compactness to prove that the image of a compact metric space under a continuous map is compact. 5. Show that a finite union of compact metric spaces is compact.
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Unformatted text preview: 6. Let ( X,d ) be metric space and A ⊆ X . (a) Define d ( x,A ) = inf { d ( x,a ) | a ∈ A } for x ∈ X . Prove that d ( x,A ) is a continuous function from X to R . (b) True or false: d ( x,A ) = d ( x,a ) for some a ∈ A . What about if A is closed?. . . and if A is compact? (c) True or false: x ∈ cl A ⇐⇒ d ( x,A ) = 0. (d) Define the ±-neighborhood of A in X to be the union U ( A,± ) = [ a ∈ A B ± ( a ) . Prove that U ( A,± ) = { x | d ( x,A ) < ± } . (e) Show that if A and B are disjoint closed subset of X and B is compact, then for some ± , the ±-neighborhoods of A and B are disjoint. (f) Is this true if B is not compact? 1 7. Suppose that for some ± > 0, every ±-ball in X has compact closure. Show that X is complete. 8. Suppose that for each x ∈ X there is an ± > 0 such that B ± ( x ) has compact closure. Show by means of an example that X need not be complete. 2...
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  • Fall '08
  • Staff
  • Economics, Topology, Metric space, compact metric space, compact metric spaces, open cover definition

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