Difference between revisions of "Closure"
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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 ''A''. It is a [[function]] ''cl'' from ''A'' to the [[power set]] of ''A'' satisfying the following conditions: | 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> ( | + | * <math>A\in cl(A)</math> ([[augment]]ation) |
| − | * If <math>B\subset A</math> then <math>cl(B)\subset cl(A)</math> (monotonicity) | + | * If <math>B\subset A</math> then <math>cl(B)\subset cl(A)</math> ([[Monotone|monotonicity]]) |
| − | * <math>cl(A)\in cl(A)</math> (idempotence) | + | * <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. | In the category of [[topological space]]s, this operator is isomorphic to the standard topological one. | ||
[[Category:Topology]] | [[Category:Topology]] | ||
Revision as of 03:34, June 28, 2008
In topology, the closure of a set A is the intersection of all closed sets containing A. In metric spaces, this can be also defined as the set of all limit points 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 spaces, this operator is isomorphic to the standard topological one.