For any set S ⊆ ℝ, let denote the intersection of all the closed sets containing S.(a) Pr...

For any set S ⊆ ℝ, let denote the intersection of all the closed sets containing S.

(a) Prove that is a closed set.

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(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.

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(c) Prove that = cl S.

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(d) If S is bounded, prove that is bounded.