Closed set
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Revision as of 15:20, 21 June 2011 by Boris Tsirelson (talk  contributions)
In mathematics, a set , where (X,O) is some topological space, is said to be closed if its complement in X, the set , is open. The empty set and the set X itself are always closed sets. The finite union and arbitrary intersection of closed sets are again closed.
[edit] Examples

Let X be the open interval (0, 1) with the usual topology induced by the Euclidean distance. Open sets are then of the form
 .

As a more interesting example, consider the function space C[a,b] (with a < b). This space consists of all realvalued continuous functions on the closed interval [a, b] and is endowed with the topology induced by the norm
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