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A ternary relational structure 〈 , [⋅,⋅,⋅]〉, interpreting a notion of betweenness, gives rise to the family of intervals, with interval [ , ] 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 { , } ↦ [ , ]. We investigate two broad scenarios: the first involves metric spaces and Menger's betweenness interpretation; the second deals with continua and the subcontinuum interpretation.

Given the lattice of all topologies definable for an infinite set X, a technique to solve many minimality problems is developed. Its potential in characterizing and, where possible, identifying those topologies that are minimal with respect to various invariants, including TA, is illustrated. Finally, an alternative description of each topologically established minimal structure in terms of the behavior of the naturally occurring specialization order and the intrinsic topology on the resulting partially ordered set is offered.

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