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.