The upper set \(\widetilde A\) of a metric space \(A\) is a subset of \(A\times(0,\infty)\), consisting of all pairs \((x,|x-y|)\) with \(x,y \in A\), \(x\neq y\). The authors consider various properties of \(\widetilde A\) and a metric of \(\widetilde A\), called the broken hyperbolic metric. The theory is applied to study basic properties of quasisymmetric maps.