Difference between revisions of "Closure (topology)"

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imported>Hendra I. Nurdin
(Stub for closure)
 
imported>Richard Pinch
m (Closure (mathematics) moved to Closure (topology): There are other meaning within mathematics)
 
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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 [[topological space|limit points]] in ''X''. It is usually denoted by <math>\overline{A}</math>. Other equivalent definitions of the closure of A is as the smallest [[closed set]] in ''X'' containing ''A'', or the intersection of all closed sets in ''X'' containing ''A''.
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[[Category:Mathematics_Workgroup]]
In [[mathematics]], the '''closure''' of a subset ''A'' of a [[topological space]] ''X'' is the set union of ''A'' and ''all'' its [[topological space#Some topological notions|limit points]] in ''X''. It is usually denoted by <math>\overline{A}</math>. 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''. 
[[Category:CZ Live]]
 
==Properties==
* A set is contained in its closure, <math>A \subseteq \overline{A}</math>.
* The closure of a closed set ''F'' is just ''F'' itself, <math>F = \overline{F}</math>.
* Closure is [[idempotence|idempotent]]: <math>\overline{\overline A} = \overline A</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.
:<math>(X - A)^{\circ} = X - \overline{A};~~ \overline{X-A} = X - A^{\circ}.</math>

Latest revision as of 15:20, 6 January 2009

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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.