On hyperspaces of generalized metric spaces (Q676969)

Let \(\mathcal K(X)\) (resp. \(\mathcal F(X)\)) denote the space of all non-empty compact (resp. finite) subsets of a (regular \(T _2\)) space \(X\) endowed with the Vietoris topology. The paper concerns the following question: If \(X\) belongs to a class of topological spaces, does \(\mathcal K(X)\) or \(\mathcal F(X)\) belong to the same class? The paper contains a table giving answers to this question for some classes of generalized metric spaces. Some comments (without proofs) concerning the table are also given.