For any set S ⊆ , let denote the intersection of all the closed sets containing S (a) Prove that is a closed set. (b) Prove that is the smallest closed set containing S. That is, show that S ⊆ ,...

For any set S ⊆
, let

denote the intersection of all the closed sets containing S

(a) Prove that

is a closed set.

(b) Prove that

is the smallest closed set containing S. That is, show that

S ⊆

, and if C is any closed set containing S, then

⊆ C.

(c) Prove that

= cl S.

(d) If S is bounded, prove that

is bounded