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