Review on: The compact-open topology: A new perspective

The paper is a contribution to the theory of function spaces. Given a Tychonoff space X the set KC(X) consists of all realvalued functions that are continuous on compact subsets. It is endowed with the compact- open topology. In the paper properties of the resulting topological space KCk(X), of its uniformity of uniform convergence on compact sets, or its locally convex structure are investigated. It is shown that for any space X the uniformity is complete. Topological properties of KCk(X) are investigated and are compared to the analogous property of its dense subspace Ck(X), consisting of all continuous functions. Among other results, characterizations are obtained for (sub)metrizability, for complete metrizability or for other types of completeness, such as Cech completeness and pseudo completeness, for separability and second countability. The main motivation for considering the larger space KCk(X) lies in the attempt of having a simpler proof for the characterization of some completeness properties for the subspace Ck(X), obtained by R.A. McCoy in Topology Appl. 22 (1986).