Editing Compactly generated space (section)

===Definition 1===

Informally, a space whose topology is determined by its compact subspaces, or equivalently in this case, by all continuous maps from arbitrary compact spaces.

A topological space <math>X</math> is called '''compactly-generated''' or a '''k-space''' if it satisfies any of the following equivalent conditions:<ref>{{cite journal |last1=Lawson |first1=J. |last2=Madison |first2=B. |title=Quotients of k-semigroups |journal=Semigroup Forum |date=1974 |volume=9 |pages=1–18 |doi=10.1007/BF02194829}}</ref>{{sfn|Willard|2004|loc=Definition 43.8}}{{sfn|Munkres|2000|p=283}}

:(1) The topology on <math>X</math> is [[coherent (topology)|coherent]] with the family of its compact 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 subspace <math>K\subseteq X.</math>
:(2) 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 spaces <math>K.</math>
:(3) <math>X</math> is a [[quotient space (topology)|quotient space]] of a [[topological sum]] of compact spaces.
:(4) <math>X</math> is a quotient space of a [[weakly locally compact]] space.

As explained in the [[final topology]] article, condition (2) is well-defined, even though the family of continuous maps from arbitrary compact spaces is not a set but a proper class.

The equivalence between conditions (1) and (2) follows from the fact that every inclusion from a subspace is a continuous map; and on the other hand, every continuous map <math>f:K\to X</math> from a compact space <math>K</math> has a compact image <math>f(K)</math> and thus factors through the inclusion of the compact subspace <math>f(K)</math> into <math>X.</math>