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

[wlawvere@buffalo.edu]

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:
> > 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.