Motivation
One of the primary motivations for studying compactly generated spaces comes from category theory. The category of topological spaces, Top, is defective in the sense that it fails to be a cartesian closed category. There have been various attempts to remedy this situation, one of which is to restrict oneself to the full subcategory of compactly generated Hausdorff spaces. This category is, in fact, cartesian closed. A definition of the exponential object is given below.
These ideas can be generalised to the non-Hausdorff case, see section 5.9 in the book `Topology and groupoids' listed below. This is useful since identification spaces of Hausdorff spaces need not be Hausdorff.
Read more about this topic: Compactly Generated Space
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