cogenerator of k-spaces

cogenerator of k-spaces

[Gaucher Philippe]

Dear All,

Does the category of k-spaces (i.e. colimits of compact Hausdorff spaces) have
a cogenerator ? Note that these spaces are not necessarily normal because
they are not necessarily Hausdorff. Otherwise [0,1] would have work without
any additional argument.

Thanks in advance. pg.

Re: cogenerator of k-spaces

[Gabor Lukacs]

Dear Philippe,

You wrote "colimits of compact Hausdorff spaces." In what category? There
is both a Hausdorff and a non-Hausdorff notion for k-spaces. The earlier
is certainly easier to define.

You may wish to look at Chapter 1 of my thesis:

http://at.yorku.ca/p/a/a/o/41.htm

I discuss there both notions, and provide some reference to the
literature, so it might be a good starting point. Spaces where points can
be distinguished by continuous real-valued functions are often referred to
as "functionally Hausdorff."

T. Ishii gives an example of a regular k-space that is not functionally
Hausdorff in his paper "On the Tychonoff functor and $w$-compactness" that
appeared in Topology Appl., 11(2):173--187, 1980. (See Lemma 3.1.) (I can
send you the paper if you are interested.) This, of course, means that
even for Hausdorff k-spaces you cannot use [0,1] as a cogenerator.

Now, as I am writing this, I wonder what you mean by 'cogenerator': An
object such that morphisms into that object can distinguish between
morphisms in the category (i.e., generalized points), or something that
every space in your category admits an __embedding__ to some power of this
object?

The notion of "embedding" already requires some kind of factorization
system, though.

For the earlier, however, {0,1} equipped with the anti-discrete topology
(i.e., only the set itself and the emptyset are open) does what you want.
It is certainly a k-space (albeit non-Hausdorff).


Best wishes,
Gabi



On Fri, 1 Feb 2008, Gaucher Philippe wrote:

> Dear All,
>
> Does the category of k-spaces (i.e. colimits of compact Hausdorff spaces) have
> a cogenerator ? Note that these spaces are not necessarily normal because
> they are not necessarily Hausdorff. Otherwise [0,1] would have work without
> any additional argument.
>
> Thanks in advance. pg.
>