Set-maximal selections

The present paper deals with continuous selections f for the Vietoris hyperspace of all nonempty closed subsets of a space X which are maximal with respect to some closed subset SX, i.e. with the property that f(T)S for every 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.