Range universal spaces (Q664686)

The topological spaces \(Y\) are characterized for which the Isbell and finest splitting topologies on the set \(C(X,Y)\) of all continuous functions from \(X\) to \(Y\) coincide for all topological spaces \(X\). Such spaces are, after identifying points that are in each others closures, subsets of the two point Sierpiński space. The same question is considered for the coincidence of the restriction of the finest splitting topology on the upper semicontinuous set-valued functions to \(C(X,Y)\) and the finest splitting topology on \(C(X,Y)\). These spaces are, after the same identification, order bases for bounded complete continuous directed complete partial orders with the Scott topology.