My question is only concerning part (vi) of the question.
Sorry for posting the whole question, but I thought it might help someone to understand the context in which the question arises.
I got up to part (vi). I was wondering if the right track was to define , take the intersection of all the , and prove that this new set is closed. If yes, I did not succeed, a little hint would be appreciated.
Let X be a metric space and let E be a nonempty subset of X. ( ) Define .
(i) Use the triangle inequality to show that .
(ii) Switching the roles of x and x′ and combining, show that .
(iii) Deduce that is a continuous function.
(iv) If in addition E is a closed subset of X, show that .
(v) Deduce from (iii) that for fixed t > 0, is an open subset of X containing E.
(vi) Choosing , in (v) show that every closed set of X can be written as the intersection of countably many open subsets of X.