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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