Editing Compactly generated space (section)

===Definition 3===

Informally, a space whose topology is determined by its compact Hausdorff subspaces.

A topological space <math>X</math> is called '''compactly-generated''' or a '''k-space''' if its topology is [[coherent (topology)|coherent]] with the family of its compact Hausdorff subspaces; namely, it satisfies the property:
:a set <math>A\subseteq X</math> is open (resp. closed) in <math>X</math> exactly when the intersection <math>A\cap K</math> is open (resp. closed) in <math>K</math> for every compact Hausdorff subspace <math>K\subseteq X.</math>

Every space satisfying Definition 3 also satisfies Definition 2.  The converse is not true.  For example, the [[Sierpiński space]] <math>X=\{0,1\}</math> with topology <math>\{\emptyset,\{1\},X\}</math> does not satisfy Definition 3, because its compact Hausdorff subspaces are the singletons <math>\{0\}</math> and <math>\{1\}</math>, and the coherent topology they induce would be the [[discrete topology]] instead.  On the other hand, it satisfies Definition 2 because it is [[homeomorphic]] to the quotient space of the compact interval <math>[0,1]</math> obtained by identifying all the points in <math>(0,1].</math>

By itself, Definition 3 is not quite as useful as the other two definitions as it lacks some of the properties implied by the others.  For example, every quotient space of a space satisfying Definition 1 or Definition 2 is a space of the same kind.  But that does not hold for Definition 3.

However, for [[weak Hausdorff]] spaces Definitions 2 and 3 are equivalent.{{sfn|Strickland|2009|loc=Lemma 1.4(c)}}  Thus the category [[Category of compactly generated weak Hausdorff spaces|CGWH]] can also be defined by pairing the weak Hausdorff property with Definition 3, which may be easier to state and work with than Definition 2.