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 T 1 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.