It is interesting that an answer to this question, before it was asked here in the categories list, came up two weeks ago in a discussion I had with Jimmie Lawson and Matthias Schroeder. Schroeder showed recently that N^(N^N), where N is discrete and the exponential is calculated in k-spaces, is not regular, and hence not zero-dimensional either, which was an open problem (notice that the compact-open topology on N^(N^N) is easily seen to be zero-dimensional). Lawson observed that this is isomorphic to Z^(N^N), which, with the pointwise operations, is an abelian group in the category of k-spaces. This gives your counter-example. Lawson also said that counter-examples to complete regularity of k-groups where previously known among the experts in the subject, but were more complicated and/or artificial. (I don't know references.) I hope this helps. Martin Escardo Michael Barr writes:

I have not thought deeply on this, but it strikes me that the basic problem is that such a group might not have a uniform topology. Such a group will have, I think, a separately continuous multiplication and hence, if U is a neighborhood of the identity {xU|x \in G} will be a cover, but there would seem no obvious reason for it to have a *-refinement. A continuous homomorphism would be uniformly continuous for those covers, if they do form a uniformity, it seems to me.

If only John Isbell were still around to answer this kind of question, a wish I have wished many times since and well before his demise. But have you looked in his uniform spaces book? That is the sort of thing he might well have considered. If I were around the math library, I would look.

Michael

On Thu, 25 Sep 2008, Bill Rowan wrote:

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Hi all,

Does anyone know of a good place where someone has written down the basic properties of such objects? As an example, if we have an (abelian, say) topological group, there is a natural uniform topology on the group such that the operations are uniformly continuous. Does the same hold for abelian group objects in the category of Kelley spaces? But anything would be helpful.

Bill Rowan

The term "Kelley space" is a misnomer (due to Gabriel & Zisman ?), resulting from a misinterpretation of the prefix in "k-space". JLK's excellent 1955 textbook, from which many of us learned mathematics, was not doing subtle self-promotion when he used that term in his clear exposition. In fact, "k" stands for kompakt, and the term was used by Hurewicz in his 1949 lectures at Princeton where he introduced these spaces. I have that from telephone discussions with the late David Gale, who had mentioned Hurewicz's k-spaces in his 1950 PAMS paper (as noticed by Horst Herrlich). The same implicit idea is used in RH Fox's 1945 paper (except based on countable compact spaces instead of all), which was directly incited by a letter from Hurewicz. Bill On Mon 09/29/08 6:36 AM , Jeff Egger jeffegger@yahoo.ca sent:

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if we have an (abelian, say) topological group, there is a natural uniform topology on> the group such that the operations are uniformly continuous. Does the> same hold for abelian group objects in the category of Kelley spaces? As others have already noted, the answer is no. One possible solution (assuming you regard this as a defect) is to apply the idea implicit in the definition of Kelley space, not to the category of all topological spaces, but to that of all Tychonov (=uniformisable) spaces. What results is a cartesian closed category (that of "k_R-Tychonov spaces") with somewhat different properties; a group in this category is tautologously uniformisable and, if I recall correctly, is also true that the operations are uniformly continuous. Gabor Lukacs has studied these things and spoken about them at several conferences.

Cheers, Jeff.