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.