Editing Closed set (section)

== More about closed sets ==

The notion of closed set is defined above in terms of [[open set]]s, a concept that makes sense for [[topological space]]s, as well as for other spaces that carry topological structures, such as [[metric space]]s, [[differentiable manifold]]s, [[uniform space]]s, and [[gauge space]]s. 

Whether a set is closed depends on the space in which it is embedded. However, the [[Compact space|compact]] [[Hausdorff space]]s are "[[H-closed space|absolutely closed]]", in the sense that, if you embed a compact Hausdorff space <math>D</math> in an arbitrary Hausdorff space <math>X,</math> then <math>D</math> will always be a closed subset of <math>X</math>; the "surrounding space" does not matter here. [[Stone–Čech compactification]], a process that turns a [[Completely regular space|completely regular]] Hausdorff space into a compact Hausdorff space, may be described as adjoining limits of certain nonconvergent nets to the space.

Furthermore, every closed subset of a compact space is compact, and every compact subspace of a Hausdorff space is closed.

Closed sets also give a useful characterization of compactness: a topological space <math>X</math> is compact if and only if every collection of nonempty closed subsets of <math>X</math> with empty intersection admits a finite subcollection with empty intersection.

A topological space <math>X</math> is [[Disconnected space|disconnected]] if there exist disjoint, nonempty, open subsets <math>A</math> and <math>B</math> of <math>X</math> whose union is <math>X.</math> Furthermore, <math>X</math> is [[totally disconnected]] if it has an [[open basis]] consisting of closed sets.