Convexity in topological betweenness structures (Q2665203)

A betweenness structure is a pair \(\langle X,[\cdot,\cdot,\cdot] \rangle\), where \(X\) is a set and \([\cdot,\cdot,\cdot]\subset X^{3}\) is a ternary relation satisfying that \begin{itemize} \item[(B1)] Inclusivity: \((\forall\ xy )\) \(([x,y,y] \wedge [x,x,y])\) \item[(B2)] Symmetry: \((\forall\ xzy )\) \(([x,z,y] \rightarrow [y,z,x])\) \item[(B3)] Uniqueness: \((\forall\ xz)\) \(([x,z,x]\rightarrow x=z)\) \end{itemize} Given a betweenness structure \(\langle X,[\cdot,\cdot,\cdot] \rangle\), an interval is defined as \([a,b]=\{c\in X:[a,c,b]\}\), a convex subset of \(X\) is a subset \(C\) of \(X\) such that \(a,b\in C\), implies \([a,b]\subset C\). The span of a subset \(A\) of \(X\) is defined as \([A]=\bigcup\{[a,b]:a,b\in A\}\). The convex hull of \(A\) is defined as \([A]^{\omega}=\bigcup\{[A]^{n}:n\in\omega\}\), where \([A]^{0}=A\) and \([A]^{n+1}=[[A]^{n}]\). With a detailed analysis of examples, in this paper the authors show how the notion of betweenness is related to several important concepts in mathematics. In particular if besides a betweenness structure, \(X\) has a topology, it is possible to define interesting relations between the two structures, starting by asking that intervals are closed. In this sense, the authors define local convexity, upper (and lower) semi-continuity of betweenness, and a type of internal continuity of the betweenness. They obtain results connecting the convexity and the topology in compact connected Hausdorff spaces which are aposyndetic or hereditary unicoherent. In particular, they study how the span and the convex hull interact with the topological closure and interior operators.