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Let $X$, $Y$, and $Z$ be locally-compact, complete, and separable metric spaces and suppose that $X$ is compact; all non-empty.

Consider the spaces $C(X,C(Y,Z))$ and $C(X\times Y,Z)$ both equipped with their uniform convergence on compact topologies. Does the map: $$ C(X\times Y,Z)\ni f \to (x\mapsto [y\mapsto f(x,y)]) \in C(X,C(Y,Z)), $$ define a homomorphism; and if so, is it uniformly continuous?

References?: Where can I read more about this map...surely this has been studied somewhere in the general topology literature; probably in the mid 1950-1960s I expect?

SetValued_Michael
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    I don't have my copy with me, but I would look at Engelking's General Topology, or Gillman & Jerrison's Rings of Continuous Functions, for something like this. – Iian Smythe Jul 06 '21 at 19:00
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    Reference: https://ncatlab.org/nlab/show/exponential+law+for+spaces – Robert Furber Jul 06 '21 at 19:19
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    Locally compact spaces always work. But then the exponential of one locally compact space with respect to another is not usually locally compact. "Convenient categories" of topological spaces are specially defined so that this isomorphism exists, for a suitable choice of topology on the continuous functions, but the product used doesn't agree with the usual product of topological spaces except in certain special cases (which depend on the convenient category used). Common examples are sequential spaces and compactly generated spaces. – Robert Furber Jul 06 '21 at 19:26
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    @RobertFurber Building on that point; this article: An Elementary Approach to Exponential Spaces by G. Richter and E. Lower-Colebnders (see: https://link.springer.com/content/pdf/10.1023/A:1011268007097.pdf ) gives a nice sufficient condition that a space be quasi-locally compact (as opposed to the usual condition of Hausdorff + Locally-compact)...note, here I use the coincides between the uniform convergence on compacts and compact-open topologies. – ABIM Jul 06 '21 at 19:30
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    If memory serves, Kelley in his famous topology book does look at uniform function spaces. – Todd Trimble Jul 06 '21 at 19:56

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In the context of functions from one space to another, a key-word here might by "currying", for the process of converting $x,y\to f(x,y)$ to $x\to (y\to f(x,y)$. In the case of spaces of holomorphic functions with the natural topology, verification is not hard.

In other contexts, isomorphisms $\operatorname{Hom}(X\otimes Y,Z)\approx \operatorname{Hom}(X,\operatorname{Hom}(Y,Z))$ are instances of the Eilenberg-Maclane adjunction. In simple algebraic situations (e.g., for abelian groups) the verification is easy.

Those instances suggest that we might want to characterize the various topologies in a fashion so as to make the assertion be correct... unless there is a "counter-example" that is important not to exclude for your purposes. That is, in effect, do you need/want the isomorphism to be correct, or not necessarily. :)

SetValued_Michael
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paul garrett
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  • Yes! I also felt this was some sort of Hom-Tensor adjuction (reminiscent of the one I known of...in $_RMod$) but does the product $\times$ behave as some sort of tensor in $Top$? – SetValued_Michael Jul 06 '21 at 19:13
  • @SetValued_Michael, I think Cartesian product of topological spaces cannot be a tensor product in a literal sense, ... after all, it's hard to say the relevant things in the cat of top spaces and continuous maps... but perhaps abstracting the diagrammatic characterization would be illuminating, if not definitive. Dunno. – paul garrett Jul 06 '21 at 19:17
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    Another keyword would be exponential law for spaces: https://ncatlab.org/nlab/show/exponential+law+for+spaces – M.G. Jul 06 '21 at 19:19
  • @M.G., that looks like it should/could answer some form of this question! :) – paul garrett Jul 06 '21 at 19:20
  • This is all very helpful, but I still do wonder if this ''adjunction map'' is uniformly continuous (if all considered spaces are locally-compact metric spaces; thus locally-compact + Hausdorff as in the above comments) – SetValued_Michael Jul 06 '21 at 19:33
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    Things get tricky for uniform spaces (such as your locally-compact metric spaces); (see: https://mathoverflow.net/questions/95923/category-of-uniform-spaces); I just note this since some care should be taken in this situation :) – ABIM Jul 06 '21 at 19:37
  • @Anastasis Thanks I'll take a look, the references all seem very technical; which likely is a good thing. – SetValued_Michael Jul 06 '21 at 19:43