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

Authors:
Hideki Tsuiki;Yasunao Hattori
Affiliations:
Graduate School of Human and Environmental Studies, Kyoto University, Kyoto, 606-8501, Japan;Department of Mathematics, Shimane University, Matsue, Shimane, 690-8504, Japan
Venue:
Theoretical Computer Science
Year:
2008

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Abstract

Let (X,d) be a metric space and BX=XxR denote the partially ordered set of (generalized) formal balls in X. We investigate the topological structures of BX, 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 BX.