Difference between revisions of "Closure"
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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> | |
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| + | 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: | ||
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| + | * <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]]) | ||
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| + | In the category of [[topological space]]s, this operator is isomorphic to the standard topological one. | ||
[[Category:Topology]] | [[Category:Topology]] | ||
Latest revision as of 04:19, January 7, 2017
In topology, the closure of a set is the intersection of all closed sets containing . Equivalently, the closure of is the union of and all limit points of
A closure operator is an abstract (category theory) form of the topological notion of closure which can be applied to any set . It is a function from to the power set of satisfying the following conditions:
- (augmentation)
- If then (monotonicity)
- (idempotence)
In the category of topological spaces, this operator is isomorphic to the standard topological one.