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Exercise 4.11. For each of the following, state whether it is true or false. If true,...

Exercise 4.11. For each of the following, state whether it is true or false. If true, prove. If false, provide a counterexample.

  1. (i) LetX andY besetsfromRn. IfX⊂Y thenX is closed if and only if Y is closed.

  2. (ii) Let X and Y be sets from Rn. If X ∩Y is closed and convex then eitherX or Y is closed and convex (one or the other).

  3. (iii) LetX beanopensetandY ⊆X. IfY ≠∅,thenY isaconvexset.

  4. (iv) SupposeX isanopensetandY isaconvexset. IfX∩Y ⊂X then

    X∪Y is open.

  5. (v) If AandB are closed sets,and A∩B=∅ then A∪B is a closed set.

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