Editing Compactly generated space (section)

==Motivation==

Compactly generated spaces were originally called '''k-spaces''', after the German word ''kompakt''. They were studied by [[Hurewicz]], and can be found in General Topology by Kelley, Topology by Dugundji, Rational Homotopy Theory by Félix, Halperin, and Thomas.

The motivation for their deeper study came in the 1960s from well known deficiencies of the usual [[category of topological spaces]]. This fails to be a [[cartesian closed category]], the usual [[cartesian product]] of [[identification map]]s is not always an identification map, and the usual product of [[CW-complex]]es need not be a CW-complex.<ref name=hatcher>{{cite book|last=Hatcher|first=Allen|url=https://pi.math.cornell.edu/~hatcher/AT/AT.pdf |title=Algebraic Topology|year=2001}} (''See the Appendix'')</ref> By contrast, the category of simplicial sets had many convenient properties, including being cartesian closed. The history of the study of repairing this situation is given in the article on the [[nLab|''n''Lab]] on [http://ncatlab.org/nlab/show/convenient+category+of+topological+spaces convenient categories of spaces].

The first suggestion (1962) to remedy this situation was  to restrict oneself to the [[full subcategory]] of compactly generated Hausdorff spaces, which is in fact cartesian closed. These ideas extend on the [[de Vries duality theorem]]. A definition of the [[exponential object]] is given below. Another suggestion (1964) was to consider the usual Hausdorff spaces but use functions continuous on compact subsets.

These ideas generalize to the non-Hausdorff case;{{sfn|Brown|2006|loc=section 5.9}} i.e. with a different definition of compactly generated spaces. This is useful since [[identification space]]s of Hausdorff spaces need not be Hausdorff.<ref>{{cite journal |last1=Booth |first1=Peter |last2=Tillotson |first2=J. |title=Monoidal closed, Cartesian closed and convenient categories of topological spaces |journal=Pacific Journal of Mathematics |date=1980 |volume=88 |issue=1 |pages=35–53 |doi=10.2140/pjm.1980.88.35 |url=https://msp.org/pjm/1980/88-1/pjm-v88-n1-p03-s.pdf}}</ref>

In modern-day [[algebraic topology]], this property is most commonly coupled with the [[weak Hausdorff]] property, so that one works in the [[Category of compactly generated weak Hausdorff spaces|category CGWH of compactly generated weak Hausdorff spaces]].