For any set S ⊆ , let S o denote the union of all the open sets contained in S. (a) Prove that S o is an open set. (b) Prove that S o is the largest open set contained in S. That is, show that S o ⊆...

For any set S ⊆
, let S^o
denote the union of all the open sets contained in S.

(a) Prove that S^o
is an open set.

(b) Prove that S^o
is the largest open set contained in S. That is, show that S

o ⊆ S, and if U is any open set contained in S, then U ⊆ S^o.

(c) Prove that S^o
= int S.