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Texas A&M University Department of Mathematics Volodymyr Nekrashevych Fall 2015 Math 636 — Problem Set 3 Issued: 09.18 Due: 09.25 3.1. Let C[0, 1] be the metric space of all continuous functions f : [0, 1] −→ R with the metric d(f, g) = maxx∈[0,1] |f (x) − g(x)|. Is the closed unit ball {f ∈ C[0, 1] : d(f, 0) ≤ 1}, where 0 is the constant zero function, compact? Prove. 3.2. Let (X, d) be a compact metric space. Show that it is complete, i.e., if a sequence (xn ) is such that for every > 0 there exists n0 such that d(xn , xm ) < for all n, m ≥ n0 , then xn converges. 3.3. Let (X, d) be a compact metric space. Show that X is separable (i.e., has a countable dense subset). 3.4. Let (X, d) be a compact metric space such that for every x, y ∈ X and every > 0 there exists a sequence x0 = x, x1 , x2 , . . . , xn = y such that d(xi , xi+1 ) < for all i = 0, 1, . . . , n − 1. Show that X is connected. (Compare with Problem 2.4.) 3.5. Prove that every countable non-empty Hausdorff compact space has an isolated point. 3.6. Let S be a well ordered set (i.e., an ordered set such that every subset has smallest element) with the order topology. Show that if S contains a maximal element then S is compact. Order topology on a linearly ordered set S is defined by the basis of open sets of the form (α, β) = {x ∈ S : α x y}, (−∞, α) = {x ∈ S : x α}, and (α, +∞) = {x ∈ S : x α}.