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**ElieWiesel** Define the $\displaystyle \epsilon $ neighborhood of a set A by: $\displaystyle N_{\epsilon} (A) = \{ x \in M : \exists y \in A $ such that $\displaystyle d(x,y) < \epsilon \} $ (that is, it is the collection of points in M wich are with $\displaystyle \epsilon $ of some point in A.

(a) prove $\displaystyle N_{\epsilon} (A) $ is open.

(b) $\displaystyle \bigcap_{\epsilon > 0} N_{\epsilon} (A) = \bar{A} $, the closure of A

(c) A subset B $\displaystyle \subset $ M is said to be a $\displaystyle G_{\delta} $ if it is the countable intersection of open sets. Modify part (b) to prove: every closed set is a $\displaystyle G_{\delta} $