Difference between revisions of "Closure (topology)"

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imported>Richard Pinch
(→‎Properties: toplogical closure operator, ie distributes over finite union)
imported>Richard Pinch
(→‎Properties: closure/interior in symbols)
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* Closure [[distributivity|distributes]] over finite [[union]]: <math>\overline{A \cup B} = \overline A \cup \overline B</math>.  
* Closure [[distributivity|distributes]] over finite [[union]]: <math>\overline{A \cup B} = \overline A \cup \overline B</math>.  
* The complement of the closure of a set in ''X'' is the [[interior (topology)|interior]] of the complement of that set; the complement of the interior of a set in ''X'' is the closure of the complement of that set.
* The complement of the closure of a set in ''X'' is the [[interior (topology)|interior]] of the complement of that set; the complement of the interior of a set in ''X'' is the closure of the complement of that set.
:<math>(X - A)^{\circ} = X - \overline{A};~~ \overline{X-A} = X - A^{\circ}.</math>

Revision as of 15:43, 27 December 2008

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In mathematics, the closure of a subset A of a topological space X is the set union of A and all its limit points in X. It is usually denoted by . Other equivalent definitions of the closure of A are as the smallest closed set in X containing A, or the intersection of all closed sets in X containing A.

Properties

  • A set is contained in its closure, .
  • The closure of a closed set F is just F itself, .
  • Closure is idempotent: .
  • Closure distributes over finite union: .
  • The complement of the closure of a set in X is the interior of the complement of that set; the complement of the interior of a set in X is the closure of the complement of that set.