Closure (topology): Difference between revisions
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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''. | 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>[[Category:Suggestion Bot Tag]] | |||
Latest revision as of 16:01, 29 July 2024
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.
