In response to Ronnie Brown's inquiries about "embeddability" and cartesian closed categories, the following three publications may be of interest: F.W. Lawvere Volterra's Functionals and the Covariant Cohesion of Space Rendiconti del Circolo Matematico di Palermo (2) Supplemento No. 64, 2000, pp 201-204. This paper is partly about the history of the problem with which Ronnie is concerned, but I only later became aware of the significance of the following two papers: Ralph H. Fox On Topologies for Function Spaces Bulletin of the American Mathematical Society, vol. 51, 1945, pp 429-432 This paper is often cited, but note that it states explicitly that it was written in response to a question in a letter by W. Hurewicz. David Gale Compact Sets of Functions and Function Rings Proceedings of the American Mathematical Society, vol. 1, 1950, pp 303-308 Here David Gale states that the definition of k-space was due to W. Hurewicz. Thus it appears that both the statement of the problem, as well as its standard solution were given by W. Hurewicz. The relevance to homotopy theory as well as to functional analysis was recognized over fifty years ago. There are actually many similar categories; an axiomatic approach (rather than a pragmatic one) is required in order to systematize the relation between them. They can be "normalized", as Peter Johnstone did for the sequential case, to become toposes; this should clarify the comparisons as well as provide categories with much more "convenient" exactness properties. Bill Lawvere ************************************************************ F. William Lawvere Mathematics Department, State University of New York 244 Mathematics Building, Buffalo, N.Y. 14260-2900 USA Tel. 716-645-6284 HOMEPAGE: http://www.acsu.buffalo.edu/~wlawvere ************************************************************