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

[Martin Escardo <m.escardo@cs.bham.ac.uk>]

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:
 >
 > > 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
 > >
 > >
 >
 >