On the greatest splitting topology
Let \(Y,Z\) be topological spaces and let \(C(Y,Z)\) denote the set of continuous mappings from \(Y\) into \(Z\). It is well known that the intersection of all admissible topologies on \(C(Y,Z)\) is the greatest splitting topology \(t\). If \(Y\) is regular and locally compact, then \(t\) is the compact open topology which is also admissible. Let \(Y\) have topology \(\tau\) and define the semi-regular reduction of \(Y\) to be the topology \(\tau_{sr}\) with base \(\{\text{int(Cl}(U)):U \in\tau\}\). We say that two spaces \((Y,\tau^0)\) and \((Y,\tau^1)\) are sr-reduction equivalent in case \(\tau^0_{sr}=\tau^1_{sr}\). In the present paper the authors show that if \(Y\) is sr-reduction equivalent to a regular locally compact space, and \(Z\) is regular, then the greatest splitting topology on \(C(Y,Z)\) is admissible. They then show how these hypotheses can be modified in various ways to obtain further results. A number of counter-examples and open questions are also given.
scientific article; zbMATH DE number 5378343
scientific article