G. Di Maio

Topologies on the space of continuous functions

Abstract Let X and Y be Tychonoff spaces and C(X, Y) be the space of all continuous functions from X to Y. The coincidence of the fine topology with other function space topologies on C(X, Y) is discussed. Also cardinal invariants of the fine topology on C(X, R ) , where R is the space of reals, are studied. To answer some questions of Di Maio and Naimpally (1992) other function space topologies are investigated, namely, Krikorian topology, open-cover topology, graph topology, topology of uniform convergence, proximal graph topology.

Uniformizing (proximal) Δ-topologies

Abstract Beer and Tamaki investigated necessary and sufficient conditions for the uniformizability of (proximal) Δ -topologies. Their proofs involved construction of special Urysohn functions. In this paper we attack the same problem using as a useful tool a uniform topology with reference to a Hausdorff uniformity patterned after the one related to the Attouch–Wets topology. We also study ΔU -topologies, proximal ΔU -topologies which are natural generalizations of the U -topology discovered by Costantini and Vitolo.

Properties related to first countability and countable compactness in hyperspaces: a new approach

Abstract We study properties related to first countability and countable compactness of uniformizable Δ -hit-and-miss hyperspace topologies. We show that they are proximal hyperspace topologies. We use the Smirnov compactification as a tool in our investigation.