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Line 1: − The '''closure''' of a set ''A'' is the intersection of all [[closed set]]s containing ''A''. In metric spaces, this can be also defined as the set of all limit points of the set ''A''.+ + + + + + + + +
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In [[topology]], the '''closure''' of a set <math>A</math> is the intersection of all [[closed set]]s containing <math>A</math>. Equivalently, the closure of <math>A</math> is the union of <math>A</math> and all [[limit point]]s of <math>A</math>
A '''closure operator''' is an abstract ([[category theory]]) form of the topological notion of closure which can be applied to any set <math>A</math>. It is a [[function]] <math>cl</math> from <math>A</math> to the [[power set]] of <math>A</math> satisfying the following conditions:
* <math>A\in cl(A)</math> ([[augment]]ation)
* If <math>B\subset A</math> then <math>cl(B)\subset cl(A)</math> ([[Monotone|monotonicity]])
* <math>cl(A)\in cl(A)</math> ([[idempotent|idempotence]])
In the category of [[topological space]]s, this operator is isomorphic to the standard topological one.
[[Category:Topology]]
[[Category:Topology]]