Editing Compactly generated space (section)

===Definition  2===

Informally, a space whose topology is determined by all continuous maps from arbitrary compact Hausdorff spaces.

A topological space <math>X</math> is called '''compactly-generated''' or a '''k-space''' if it satisfies any of the following equivalent conditions:{{sfn|Brown|2006|p=182}}{{sfn|Strickland|2009}}<ref>{{nlab|id=compactly+generated+topological+space |title=compactly generated topological space}}</ref>

:(1) The topology on <math>X</math> coincides with the [[final topology]] with respect to the family of all continuous maps <math>f:K\to X</math> from all compact Hausdorff spaces <math>K.</math>  In other words, it satisfies the condition:
::a set <math>A\subseteq X</math> is open (resp. closed) in <math>X</math> exactly when <math>f^{-1}(A)</math> is open (resp. closed) in <math>K</math> for every compact Hausdorff space <math>K</math> and every continuous map <math>f:K\to X.</math>
:(2) <math>X</math> is a quotient space of a [[topological sum]] of compact Hausdorff spaces.
:(3) <math>X</math> is a quotient space of a [[locally compact Hausdorff]] space.

As explained in the [[final topology]] article, condition (1) is well-defined, even though the family of continuous maps from arbitrary compact Hausdorff spaces is not a set but a proper class.{{sfn|Brown|2006|p=182}}

Every space satisfying Definition 2 also satisfies Definition 1.  The converse is not true.  For example, the [[one-point compactification]] of the [[Arens-Fort space]] is compact and hence satisfies Definition 1, but it does not satisfies Definition 2.

Definition 2 is the one more commonly used in algebraic topology.  This definition is often paired with the [[weak Hausdorff]] property to form the [[Category of compactly generated weak Hausdorff spaces|category CGWH of compactly generated weak Hausdorff spaces]].