Posted by J. Holmer on October 28, 2007, 8:39 pm, in reply to "Definition of Perfect" A set is perfect if it equals the set of its limit points. A perfect set is necessarily closed, but there are closed sets that are not perfect (like single point sets in R).
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Recall that a point of closure of a set E in X is a point x_0\in X such that every open set containing x_0 intersects E. A limit point, on the other hand, is point x_0\in X such that every open set containing x_0 intersects E in a point other than x_0. (we could also say x_0 intersects E\{x_0}). For example, let E = (0,1) union {2}. Then all points in [0,1] are limit points but 2 is not a limit point, although it is a point of closure.
Thus, there might be points in E that are not limit points of E, and there might be limit points of E that do not belong to E.
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