Scott topology, Isbell topology, and continuous convergence. (English) Zbl 0598.54005
Continuous lattices and their applications, Proc. 3rd Conf., Bremen/Ger. 1982, Lect. Notes Pure Appl. Math. 101, 251-271 (1985).
[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.
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.
Reviewer: J.D.Lawson