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A. Let (X, d) be a metric space so that for every E X and every r>0 the closed ball N,(z) = {ye X : d(y, z) < r} is com pact. Let be a homeomorphism. (1) Prove that f-+m-fn。fm for all n, m E Z. (2) Let z E X and suppose that F, {fn (z) : n E 2) is a closed subset of X Prove that F is a discrete subset of X (A subset Y C X is called discrete if for every y E Y there exists some Ty > 0 so that (y) n Y = {y}.)

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(x,d) m. me 2 (n) : nt Al c 1.e aure ont ヲ.fl z.)-→ ผู้») t/vmce d. k2

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