[For the entire collection see Zbl 0567.00010.]
The authors define and study the Isbell topology, a function space topology on the set Top(X,Y) of continuous functions from a space X to a space Y. A central theme of the paper is that this topology is the appropriate version of the compact-open topology for the general category of all (in particular, non-Hausdorff) spaces. A subbase for the Isbell topology is given by all [H,V], the set of all functions f such that \(f^{-1}(V)\) is in H, where V is an open subset of Y and H is a Scott open set in the lattice 0(X) of open subsets of X. The authors show that this topology coincides with the compact-open topology in the case that X is locally compact and more generally with the Scott topology in the case that X is core compact. Furthermore in this case the convergence structure corresponding to the Isbell topology is precisely that of continuous convergence. The authors also consider what conditions on X give rise to this topology being the topology of pointwise convergence; here one needs that X is locally finitely bottomed in the earlier sense of Isbell. Other aspects of the limitierung of continuous convergence and of the Isbell topology are also considered.