Let S and T be subsets of . Prove the following. (a) int S is an open set. (b) int (int S ) = int S (c) int (S ∩ T ) = (int S ) ∩ (int T ) (d) (int S ) ∪ (int T ) ⊆ int (S ∪ T ) (e) Find an example to...

Let S and T be subsets of
. Prove the following.

(a) int S is an open set.

(b) int (int S ) = int S

(c) int (S ∩ T ) = (int S ) ∩ (int T )

(d) (int S ) ∪ (int T ) ⊆ int (S ∪ T )

(e) Find an example to show that equality need not hold in part (d).