[For the entire collection see Zbl 0567.00010.]
The category of continuous mappings between topological spaces fails to have decently behaved function spaces, and so it is not convenient for several applications. One usually proves that the canonical bijection \(\rho\) from \({\mathcal F}(X\times Y;Z)\) to \({\mathcal F}(X\); F(X;Z)) induces a bijection \({\tilde \rho}\) from \({\mathcal C}(X\times Y;Z)\) to \({\mathcal C}(X,{\mathcal C}_ t(Y;Z))\) for topological spaces X, Y, Z in case X is separated, Y is locally compact and t is the topology of compact convergence. Although a satisfactory and successful solution to this dilemma is already known and (as S. MacLane has pointed out) can be obtained by restricting oneself to compactly generated spaces, the authors decided to investigate anew those spaces Y which are characterized by the following property: For every space Z there exists a topology t on \({\mathcal C}(Y;Z)\) such that for every space X the canonical bijection \(\rho\) induces a bijection \({\tilde \rho}\) as above.