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closure∗ mathwizard† 2013-03-21 13:02:37 The closure A of a subset A of a topological space X is the intersection of all closed sets containing A. Equivalently, A consists of A together with all limit points of A in X or equivalently x ∈ A if and only if every neighborhood of x intersects A. Sometimes the notation cl(A) is used. X If it is not clear, which topological space is used, one writes A . Note that X Y if Y is a subspace of X, then A may not be the same as A . For example, if X Y X = R, Y = (0, 1) and A = (0, 1), then A = [0, 1] while A = (0, 1). ∗ hClosurei created: h2013-03-21i by: hmathwizardi version: h31191i Privacy setting: h1i hDefinitioni h54A99i † This text is available under the Creative Commons Attribution/Share-Alike License 3.0. You can reuse this document or portions thereof only if you do so under terms that are compatible with the CC-BY-SA license. 1