Semicontinuity of Betweenness Functions

Abstract A ternary relational structure 〈 X , [ ⋅ , ⋅ , ⋅ ] 〉 , interpreting a notion of betweenness, gives rise to the family of intervals, with interval [ a , b ] being defined as the set of elements of X between a and b . Under very reasonable circumstances, X is also equipped with some topological structure, in such a way that each interval is a closed nonempty subset of X . The question then arises as to the continuity behavior—within the hyperspace context—of the betweenness function { x , y } ↦ [ x , y ] . We investigate two broad scenarios: the first involves metric spaces and Menger's betweenness interpretation; the second deals with continua and the subcontinuum interpretation.