On the topology of D-metric spaces and generation of D-metric spaces from metric spaces

Abstract

An example of a D-metric space is given, in which D-metric convergence does not define a topology and in which a convergent sequence can have infinitely many limits. Certain methods for constructing D-metric spaces from a given metric space are developed and are used in constructing (1) an example of a D-metric space in which D-metric convergence defines a topology which is T1 but not Hausdorff, and (2) an example of a D-metric space in which D-metric convergence defines a metrizable topology but the D-metric is not continuous even in a single variable.