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

[Jeff Egger <jeffegger@yahoo.ca>]

> 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.




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