Difference between revisions of "Closure"
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| − | + | 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''. | |
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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: | ||
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| + | * <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) | ||
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| + | In the category of [[topological space]]s, this operator is isomorphic to the standard topological one. | ||
[[Category:Topology]] | [[Category:Topology]] | ||
Revision as of 22:06, June 13, 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.