Topologies on hyperspaces (Q2910049)

Let \(Y, Z\) be arbitrary topological spaces and \(C(Y, Z)\) the set of all continuous functions from \(Y\) to \(Z\). Let \(\mathcal O(Z)\) stand for the set of all open subsets of \(Z\) and \(\mathcal O_Z (Y ) :=\{f^{ -1} (U ) : U \in \mathcal O(Z), f \in C(Y, Z)\}\). If \(\mathcal A\) is an arbitrary class of topological spaces, a topology \(\tau\) on \(\mathcal O_Z (Y )\) is called \(\mathcal A\)-proper if for every space \(X\) in \(\mathcal A\) the continuity of a map \(F : X \times Y \to Z\) implies the continuity with respect to the first variable of the map \(\bar F : X \times \mathcal O(Z)\to (\mathcal O_Z (Y ), \tau )\), where \(\bar F (x, U ) := F^{ -1}_{x} (U )\), \(x\) running in \(X\) and \(U\) running in \(\mathcal O(Z)\). A topology \(\tau\) on \(\mathcal O_Z (Y )\) is called \(\mathcal A\)-admissible if for every space \(X\) in \(\mathcal A\) and every map \(G : X \to C(Y, Z)\) the continuity with respect to the first variable of the map \(\bar G : X \times \mathcal O(Z)\to (\mathcal O_Z (Y ), \tau)\), where \(\bar G(x, U) := (G(x))^{-1} (U ), x \in X, U\in \mathcal O(Z)\), implies the continuity of the map \(\tilde G : X \times Y \to Z\), where \(\tilde G(x, y) := G(x)(y), x \in X, y \in Y\). A topology \(\tau\) on \(\mathcal O_Z (Y )\) is said proper or admissible when \(\mathcal A\) contains all topological spaces. Relating these notions to the classical properness and admissibility for topologies on \(C(Y, Z)\) the author gives necessary and sufficient conditions on the class \(\mathcal A\) for \(\mathcal A\)-properness (\(\mathcal A\)-admissibility) to be properness (admissibility). Furthermore, properties of the finest \(\{X\}\)-proper topology on \(\mathcal O_Z (Y )\) are studied for several metrizable spaces \(X\).