Vietoris continuous selections and disconnectedness-like properties

Abstract

Abstract. Suppose that X is a Hausdorff space such that its Vietoris hyperspace (F (X), τ V ) has a continuous selection. Do disconnectedness-like properties of X depend on the variety of continuous selections for (F (X), τ V ) and vice versa? In general, the answer is "yes" and, in some particular situations, we were also able to set proper characterizations.