Given a subset A of a topological space X, let A' be the set of limit points of A. Let à denote the closure of A: let à = (A Union A'). (In class I wrote A with a line over it instead of Ã, but I'm not sure how to do that in a web browswer.)
1. Show that à is closed.
2. Show that à is the smallest closed subset of X containing A. In other words, if C is a closed set containing A, show that C contains à also. (Equivalently, show that à is the intersection of all closed sets containing A.)
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John H. Palmieri, Department of Mathematics, University of Notre Dame, John.H.Palmieri.2@nd.edu