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Unlike the category Top of topological spaces with continuous maps as the arrows, the full subcategory of compactly generated spaces (k-spaces) is Cartesian closed.

It seems like a very nice category of spaces, since every manifold, CW complex, first countable space, compact space, and locally compact space is a k-space.

Is there any compelling reason to work in the larger category of topological spaces rather than restricting to k-spaces? For instance, are there any spaces of interest to "topology-users" such as analysts, physicists, engineers, applied mathematicians, etc. that are not k-spaces?

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    $\begingroup$ Its worth pointing out that you can also fix $\mathbf{Top}$ by generalizing; e.g. you can replace topological spaces with convergence spaces. This is, in my opinion, is the most natural definition of the word "space," if by "space" we mean "thing with an underlying set such that functions between such things are either continuous or not." $\endgroup$ – goblin Feb 19 '16 at 9:45
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I think the main idea is that of a "category which is adequate and convenient for all purposes of topology" as I wrote in the Introduction of my first, 1963, paper, available from here. That is, the idea of looking at the properties of the category of the objects one is studying is a useful one. In analysis, one may need something different. See the book "The convenient setting for global analysis" by Kriegl and Michor, available from the AMS. There is also a nice paper by Booth and Tillotson in Pacific J. Math. 1980, on monoidal closed and convenient categories. For more information on the general point see this n-lab entry.

A problem with working even in compactly generated spaces is that the category is not locally cartesian closed, and is not a topos. Thus partial maps turn up in analysis right at the beginning, for example as solutions of differential equations, but the functional analysis of partial maps is very little worked out. See this paper on the open domain case, which has been cited.

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  • $\begingroup$ Thanks for references! By the way, I love your topology book; that's where I learned about k-spaces. $\endgroup$ – ಠ_ಠ Nov 25 '14 at 16:10
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I can't think of any direct disadvantages other than the complication of the definition. But then, engineers, analysts and applied mathematicians aren't usually bothering with exponentiability of spaces anyway-although mathematical physicists might, in trying to formalize some of the "spaces" that arise in physics.

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