Cut points in metric spaces

In this note, we will define topological and virtual cut points of finite metric spaces and show that, though their definitions seem to look rather distinct, they actually coincide. More specifically, let X denote a finite set, and let D:X×X→R:(x,y)↦xy denote a metric defined on X. The tight span T(D) of D consists of all maps f∈RX for which f(x)=supy∈X(xy−f(x)) holds for all x∈X. Define a map f∈T(D) to be a topological cut point of D if T(D)−{f} is disconnected, and define it to be a virtual cut point of D if there exists a bipartition (or split) of the support supp(f) of f into two non-empty sets A and B such that ab=f(a)+f(b) holds for all points a∈A and b∈B. It will be shown that, for any given metric D, topological and virtual cut points actually coincide, i.e., a map f∈T(D) is a topological cut point of D if and only if it is a virtual cut point of D.