If A is any subset of R', let A denote the intersection of all closed sets...

  

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If A is any subset of R', let A denote the intersection of all closed sets containing A; the set A is called the closure of A. Note that A is a closed set; prove that it is the smallest closed set containing A. Prove that ACA, (A) = A (AUB)= A UB", Ø = Ø. Give an example to show that (ANB)= ANB may not hold. If A is any subset of R', let A denote the intersection of all closed sets containing A; the set A is called the closure of A. Note that A is a closed set; prove that it is the smallest closed set containing A. Prove that ACA, (A) = A (AUB)= A UB", Ø = Ø. Give an example to show that (ANB)= ANB may not hold.