Editing Closed set (section)

== Properties ==
{{See also|Kuratowski closure axioms}}

A closed set contains its own [[Boundary (topology)|boundary]]. In other words, if you are "outside" a closed set, you may move a small amount in any direction and still stay outside the set. This is also true if the boundary is the empty set, e.g. in the metric space of rational numbers, for the set of numbers of which the square is less than <math>2.</math> 

* Any [[Intersection (set theory)|intersection]] of any family of closed sets is closed (this includes intersections of infinitely many closed sets)
* The [[Union (set theory)|union]] of {{em|[[Finite set|finitely]] many}} closed sets is closed.
* The [[empty set]] is closed.
* The whole set is closed.

In fact, if given a set <math>X</math> and a collection <math>\mathbb{F} \neq \varnothing</math> of subsets of <math>X</math> such that the elements of <math>\mathbb{F}</math> have the properties listed above, then there exists a unique topology <math>\tau</math> on <math>X</math> such that the closed subsets of <math>(X, \tau)</math> are exactly those sets that belong to <math>\mathbb{F}.</math> 
The intersection property also allows one to define the [[Closure (topology)|closure]] of a set <math>A</math> in a space <math>X,</math> which is defined as the smallest closed subset of <math>X</math> that is a [[superset]] of <math>A.</math>
Specifically, the closure of <math>X</math> can be constructed as the intersection of all of these closed supersets.

Sets that can be constructed as the union of [[Countable set|countably]] many closed sets are denoted '''[[F-sigma set|F<sub>σ</sub>]]''' sets.  These sets need not be closed.