Editing Compactly generated space (section)

===Continuity of functions===

The continuous functions on compactly generated spaces are those that behave well on compact subsets.  More precisely, let <math>f:X\to Y</math> be a function from a topological space to another and suppose the domain <math>X</math> is compactly generated according to one of the definitions in this article.  Since compactly generated spaces are defined in terms of a [[final topology]], one can express the [[continuity (topology)|continuity]] of <math>f</math> in terms of the continuity of the composition of <math>f</math> with the various maps in the family used to define the final topology.  The specifics are as follows.

If <math>X</math> is CG-1, the function <math>f</math> is continuous if and only if the [[restriction (mathematics)|restriction]] <math>f\vert_K:K\to Y</math> is continuous for each compact <math>K\subseteq X.</math>{{sfn|Willard|2004|loc=Theorem 43.10}}

If <math>X</math> is CG-2, the function <math>f</math> is continuous if and only if the [[composition (functions)|composition]] <math>f\circ u:K\to Y</math> is continuous for each compact Hausdorff space <math>K</math> and continuous map <math>u:K\to X.</math>{{sfn|Strickland|2009|loc=Proposition 1.11}}

If <math>X</math> is CG-3, the function <math>f</math> is continuous if and only if the restriction <math>f\vert_K:K\to Y</math> is continuous for each compact Hausdorff <math>K\subseteq X.</math>