Convexity in topological betweenness structures

Abstract

A betweenness structure on a set X is a ternary relation [⋅,⋅,⋅]⊆X3 that captures a rudimentary notion of one point of X lying between two others. The interval[a,b] is the set of all points lying between a and b, and a subset C of X is convex if [a,b]⊆C whenever a,b∈C. The span of a set A is the union of all intervals [a,b], where a,b∈A; by iterating the span operator countably many times, we obtain the convex hull of A. The betweenness structure is topological if X carries a topology that satisfies certain compatibility conditions with respect to betweenness; in particular, intervals are closed subsets. We are guided by questions involving how the span and convex hull operators interact with the topological closure and interior operators, especially in the domains of metric spaces and of continua. With a metric space 〈X,ϱ〉, [a,c,b] holds exactly when ϱ(a,b)=ϱ(a,c)+ϱ(c,b); and one result about this betweenness structure is that the span of any compact subset is both closed and bounded. With a continuum X, [a,c,b] holds exactly when c belongs to every subcontinuum of X that contains a and b; and one result about this betweenness structure is that when the continuum is either aposyndetic or hereditarily unicoherent, the closure of a convex subset is always convex.