For instance, replacing with the set of all Hausdorff compact subspaces gives the definition of a : a space is if it is coherent with the set of all Hausdorff compact subspaces of Hausdorff-compactly generated A is a topological space whose topology is coherent with the family of all compact Hausdorff subspaces. Sometimes in the literature a compactly generated space refers to a Hausdorff-compactly generated space. In these cases compactness is often explicitly redefined at the beginning to mean both compact and Hausdorff (and quasi-compact takes the meaning of compact). In this article we make a clear separation between compactly generated spaces and Hausdorff-compactly generated spaces, since the choice affects the statement of the associated theorems. Motivation Hausdorff-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 maps is not always an identification map, and the usual product of CW-complexes need not be a CW-complex. 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 on convenient categories of spaces. The first suggestion (1962) to remedy this situation was to restrict oneself to the full subcategory of Hausdorff-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; i.e. with compactly generated spaces. This is useful since identification spaces of Hausdorff spaces need not be Hausdorff. In modern-day algebraic topology, this property is mostly commonly coupled with the weak Hausdorff property, so that one works in the category of weak-Hausdorff Hausdorff-compactly generated (WHCG) spaces. Examples and counterexamples Most topological spaces commonly studied in mathematics are (Hausdorff-)compactly generated. In the following the bracketed (Hausdorff) properties and (Hausdorff-) prefixes are meant to be applied together. Generally, if the space is Hausdorff-compactly generated, rather than just compactly generated,