Barriers in metric spaces

Defining a subset B of a connected topological space T to be a barrier (in T ) if B is connected and its complement T − B is disconnected, we will investigate barriers B in the tight span T ( D ) = { f ∈ R X : ∀ x ∈ X f ( x ) = sup y ∈ X ( D ( x , y ) − f ( y ) ) } of a metric D defined on a finite set X (endowed, as a subspace of R X , with the metric and the topology induced by the ℓ ∞ -norm) that are of the form B = B ε ( f ) ≔ { g ∈ T ( D ) : ‖ f − g ‖ ∞ ≤ ε } for some f ∈ T ( D ) and some ε ≥ 0 . In particular, we will present some conditions on f and ε which ensure that such a subset of T ( D ) is a barrier in T ( D ) . More specifically, we will show that B ε ( f ) is a barrier in T ( D ) if there exists a bipartition (or split) of the ε -support supp ε ( f ) ≔ { x ∈ X : f ( x ) > ε } of f into two non-empty sets A and B such that f ( a ) + f ( b ) ≤ a b + ε holds for all elements a ∈ A and b ∈ B while, conversely, whenever B ε ( f ) is a barrier in T ( D ) , there exists a bipartition of supp ε ( f ) into two non-empty sets A and B such that, at least, f ( a ) + f ( b ) ≤ a b + 2 ε holds for all elements a ∈ A and b ∈ B .