Re: Group and abelian group objects in the category of Kelley spaces

[Michael Barr <barr@math.mcgill.ca>]

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:

> 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
>
>