Lawson topology of the space of formal balls and the hyperbolic topology
Let \(\mathbb R_+\) denote the set of non-negative real numbers. From a metric space \((X,d)\) one can construct a partially ordered set \((B^+X,\sqsubseteq)\) where \(B^+X=X\times \mathbb R_+,\) and \((x,r)\sqsubseteq (y,s)\) if \(d(x,y)\leq r-s.\) An element of \(B^+X\) is called a formal ball in \((X,d)\). In this article the authors also consider formal balls with negative radii and study the partially ordered set \(BX=X\times\mathbb R\) of such generalized formal balls with the same order relation. In particular they investigate when the Lawson topology (defined through the order relation \(\sqsubseteq\)) and the product topology (of the metric topology and the Euclidian topology) coincide on \(B^+X\), or on \(BX.\) For instance they show that the two topologies coincide on \(BX\) if \((X,d)\) is a totally bounded metric space. They also introduce and investigate the hyperbolic topology of a metric space \((X,d)\), which is generated by the sets \(\{y:d(a,y)-d(b,y)<s\}\) for \(a,b\in X\) and \(-d(a,b)<s\), and which in general is different from the metric topology. In particular they prove that the hyperbolic topology and the metric topology coincide on \(X\) if and only if the Lawson topology and the product topology coincide on \(BX.\)
scientific article; zbMATH DE number 5354953
scientific article