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1. Let (X,d) be a separable metric space: then x has a countable dense subset D. Define = i) Let x EX and E> 0. Show that there exists a ball B € B. such that Clearly illustrate your answer on the given diagram. Make xt explicitly clear in what order things are chosen and on what each choice depends. B(x,2) ii) Now consider any nonempty open set u in X. Show that u = UB. 2 MATH 3501 (Fall 2017) Assignment 3 2. Let (X,d) be a metric space, and suppose that E CX is non-empty and totally bounded. Construct a countable dense subset D of E by considering the centres of carefully chosen balls. Justify your answer. 3. Let E be a set in a metric space (X,d) Fill in each of the blanks with one of: E Ext DE EY Exel E E\ Eigal aE E . ae E° , or if there is no answer above that will hold in every metric space, then write none for "none of these." 1\{ex every sequence that converges to = x is eventually in E every sequence lying in E that = U converges to x is constant-tailed

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