E-4 - Special Metrics

Abstract

This chapter discusses the relationship between metric and topological properties. For a metrizable space X, there are many metrics that induce the original topology of X. It is well known that a metrizable space X is separable if and only if (iff) X admits a totally bounded metric, and X is compact iff X admits a complete totally bounded metric. J. de Groot proved that a metrizable space X admits an ultrametric iff X is strongly zero-dimensional (that is, dimX equals zero, where dimX denotes the covering dimension of X). Mid sets are a geometrically intuitive concept and several topological properties can be approached through the use of mid sets. The Cantor set and the space of irrationals have the unique midset property (UMP). The UMP of a finite discrete space can be considered in terms of graph theory. A continuum having the double midset property must be a simple closed curve. Two subsets A and B of a metric space (X, ρ) are said to be congruent if there is an isometry between them. The locally finite topology is the supremum of all of Hausdorff metric topologies induced by compatible metrics on X. The u-normality is a metric property. A closed subspace of a u-normal space is u-normal.