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Semicontinuity Of Betweenness Functions, Paul Bankston, Aisling Mccluskey, Richard J. Smith
Semicontinuity Of Betweenness Functions, Paul Bankston, Aisling Mccluskey, Richard J. Smith
Mathematics, Statistics and Computer Science Faculty Research and Publications
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.