Lawson topology of the space of formal balls and the hyperbolic topology

Let (X,d) be a metric space and denote the partially ordered set of (generalized) formal balls in X. We investigate the topological structures of , in particular the relations between the Lawson topology and the product topology. We show that the Lawson topology coincides with the product topology if (X,d) is a totally bounded metric space, and show examples of spaces for which the two topologies do not coincide in the spaces of their formal balls. Then, we introduce a hyperbolic topology, which is a topology defined on a metric space other than the metric topology. We show that the hyperbolic topology and the metric topology coincide on X if and only if the Lawson topology and the product topology coincide on .