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I'm looking for a resource that covers the interplay between function and uniform spaces and k-spaces. All the texts I've seen so far cover one or two of the three, but never the full combination in sufficient generality. E.g. Willard, Kelley, Bourbaki cover function and uniform spaces, but not k-spaces (unless Bourbaki calls them something else), while e.g. tom Dieck's "Algebraic Topology" or Brown's "Topology and Groupoids" cover k-spaces, but not the other two. And it's kind of difficult to use multiple disparate texts to learn these topics, since the hypotheses vary and there are no universally accepted definitions (at least for k-spaces).

Edit: It's a bit difficult for me to put into words exactly what's lacking in the umpteen texts I've looked at, since all have different approaches and it's difficult to compare them with one another (it's one big soup in my head). Suffice to say I'm familiar with the fundamentals of uniform spaces, what I really need is how various uniformities on subsets of $\operatorname{Set}(X,Y)$ and $\operatorname{Top}(X,Y)$ work and how that relates to the compact-open topology and to k-spaces (e.g. exponential objects and the most general forms of Arzela-Ascoli). I hope this clears it up a bit more.

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  • $\begingroup$ This question is missing necessary details or it is unclear what specifically the question is asking. It should be edited to include the missing information and clarify the problem. $\endgroup$ Commented Nov 14 at 12:13
  • $\begingroup$ What do you mean by "how various uniformities on subsets of Set(X,Y) and Top(X,Y) work"? $\endgroup$ Commented Nov 14 at 16:38
  • $\begingroup$ @Jakobian I mean the properties of different kinds of uniformities on Set(X,Y) (and its subsets) and how that relates to Top(X,Y) and the co-topology. $\endgroup$ Commented Nov 14 at 16:56

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I'm looking for a resource that covers the interplay between function and uniform spaces and k-spaces. All the texts I've seen so far cover one or two of the three, but never the full combination in sufficient generality.

It's very unclear what you could mean, but let me try to answer anyway.

A good reference for topologies on $C(X, Y)$ is Topology by Dugundji, chapters XII and XIII. Dugundji assumed Hausdorff condition. Without Hausdorff assumption there is some variation on what a $k$-space is. Pi-base gives three definitions, $k_1$-space, $k_2$-space and $k_3$-space. Dugundji mostly works with compact-open topology, though there's some passing comments about when $Y$ is given a metric. The notation Dugundji uses is $Y^X$ instead of $C(X, Y)$ (as is usual for exponential objects). The relevant part here I would say is corollary 3.2, which says that if $X\to C(Y, Z)$ is continuous and $X\times Y$ is a $k$-space, then $X\times Y\to Z$ is continuous (here those unnamed maps are related by currying; the converse is always true). Another relevant part is theorem 5.3. It says that $C(X\times Y, Z)$ and $C(X, C(Y, Z))$ (or in the more pleasant notation, $Z^{X\times Y}$ and $(Z^Y)^X$) are homeomorphic whenever $Y$ is locally compact or $X\times Y$ is a $k$-space.

Dugundji also includes section IX.11 about uniform spaces although I don't recall it being complete, and he doesn't include a relationship to $C(X, Y)$ or $k$-spaces. There's different approaches to uniform spaces, you can define them using entourages, families of pseudometrics, or uniform covers. For example Uniform spaces by Isbell considers definition by uniform covers, leaving the equivalence between the definitions as an exercise, Rings of continuous functions by Gillman and Jerison considers pseudometrics, and Bourbaki's General topology considers entourages. The approach by entourages is probably most spread. This is also part of my confusion, I don't know what results would you expect. For example, are you considering $X, Y$ to be uniform spaces, and want to consider the function space $C_u(X, Y)$ of uniform functions between them instead, and obtain similar results like above but for exponential objects? I don't know any resources which would consider that.

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  • $\begingroup$ Thank you for the suggestion, I've added some context to my question, hopefully it clears it up somewhat. $\endgroup$ Commented Nov 14 at 15:29

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