Set-maximal selections

The present paper deals with continuous selections f for the Vietoris hyperspace F(X) of all nonempty closed subsets of a space X which are maximal with respect to some closed subset S⊂X, i.e. with the property that f(T)∈S for every T∈F(X) with T∩S≠∅. This gives rise to a common point of view of several extreme-like properties studied before, it also provides an useful tool in classifying disconnectedness-like properties of spaces by continuous selections for the Vietoris hyperspace.