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 XX denote a finite set, and let D:X×X→R:(x,y)↦xyD:X×X→R:(x,y)↦xy denote a metric defined on XX. The tight span   T(D)T(D) of DD consists of all maps f∈RXf∈RX for which f(x)=supy∈X(xy−f(x))f(x)=supy∈X(xy−f(x)) holds for all x∈Xx∈X. Define a map f∈T(D)f∈T(D) to be a topological   cut point of DD if T(D)−{f}T(D)−{f} is disconnected, and define it to be a virtual cut   point of DD if there exists a bipartition (or split  ) of the support supp(f) of ff into two non-empty sets AA and BB such that ab=f(a)+f(b)ab=f(a)+f(b) holds for all points a∈Aa∈A and b∈Bb∈B. It will be shown that, for any given metric DD, topological and virtual cut points actually coincide, i.e., a map f∈T(D)f∈T(D) is a topological cut point of DD if and only if it is a virtual cut point of DD.