For a topological Hausdorff
space X we study the hyperspaces 𝒫(X), 2X and 𝒞(X) of all closed subsets, all
non-empty closed subsets and all non-empty compact subsets endowed with the
convergence of sets. In this paper we shall work with the filter description of this
convergence, as defined by Choquet [2], which however is equivalent to the
topological convergence of nets of sets as defined by Frolik and Mrówka. We shall
study the relation between properties of X and properties of the spaces 𝒫(X), 2X
and 𝒞(X) such as compactness, local compactness, regularity and the topological and
pretopological character.
