# Difference between revisions of "Closure (topology)"

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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#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''. | ||

[[ | |||

==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

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.