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Properties of the closure of a set

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If A is any subset of let denote the collection of all closed sets containing A. The set is called the closure of A.Note that is a closed set, prove that it is the smallest closed set containing A.Prove the following

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The definition of the A' closure of a set A as the intersection of all closed sets containing A is examined with several important properties of A' established. The solution is presented in a PDF file

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