Editing Compactly generated space (section)

===Products===

The [[product (topology)|product]] of two compactly generated spaces need not be compactly generated, even if both spaces are Hausdorff and [[sequential space|sequential]].  For example, the space <math>X=\Reals \setminus \{1, 1/2, 1/3, \ldots\}</math> with the [[subspace topology]] from the real line is [[first countable]]; the space <math>Y=\Reals / \{1,2,3,\ldots\}</math> with the [[quotient topology]] from the real line with the positive integers identified to a point is sequential.  Both spaces are compactly generated Hausdorff, but their product <math>X\times Y</math> is not compactly generated.{{sfn|Engelking|1989|loc=Example 3.3.29}}

However, in some cases the product of two compactly generated spaces is compactly generated:
* The product of two first countable spaces is first countable, hence CG-2.
* The product of a CG-1 space and a [[locally compact]] space is CG-1.{{sfn|Lawson|Madison|1974|loc=Proposition 1.2}}  (Here, ''locally compact'' is in the sense of condition (3) in the corresponding article, namely each point has a local base of compact neighborhoods.)
* The product of a CG-2 space and a [[locally compact Hausdorff]] space is CG-2.{{sfn|Strickland|2009|loc=Proposition 2.6}}{{sfn|Rezk|2018|loc=Proposition 7.5}}

When working in a [[category (mathematics)|category]] of compactly generated spaces (like all CG-1 spaces or all CG-2 spaces), the usual [[product topology]] on <math>X\times Y</math> is not compactly generated in general, so cannot serve as a [[categorical product]].  But its k-ification <math>k(X\times Y)</math> does belong to the expected category and is the categorical product.{{sfn|Lamartin|1977|loc=Proposition 1.11}}{{sfn|Rezk|2018|loc=section 3.5}}