Metrizable ball structures

Abstract

A ball structure is a triple \((X,P,B)\), where \(X\), \(P\) are nonempty sets and, for any \(x\in X\), \(\alpha\in P\), \(B(x,\alpha)\) is a subset of \(X\), \(x\in B(x,\alpha)\), which is called a ball of radius \(\alpha\) around \(x\). We characterize up to isomorphism the ball structures related to the metric spaces of different types and groups.