We introduce generalized D-submetrizable spaces (i.e., D-submetrizable spaces of order    and study their properties. We compare these spaces with the old submetrizable spaces and sizable spaces. These sizable spaces were introduced by Fora [2], and many results in fixed point theory were obtained. Some of those results are generalizations of the well-known Banach fixed point theorem for nonmetrizable spaces.

D-metric, sizable, submetrizable.