Weak-bases and D-spaces

It is shown that certain weak-base structures on a topological space give a D-space. This solves the question by A.V. Arhangel’skii of when quotient images of metric spaces are D-spaces. A related result about symmetrizable spaces also answers a question of Arhangel’skii. Theorem. Any symmetrizable space X is a D-space (hereditarily). Hence, quotient mappings, with compact fibers, from metric spaces have a D-space image. What about quotient s-mappings? Arhangel’skii and Buzyakova have shown that spaces with a point-countable base are D-spaces so open s-images of metric spaces are already known to be D-spaces. A collection W of subsets of a sequential space X is said to be a w-system for the topology if whenever x ∈ U ⊆ X, with U open, there exists a subcollection V ⊆ W such that x ∈ T V , S V is a weak-neighborhood of x, and S V ⊆ U . Theorem. A sequential space X with a point-countable w-system is a D-space. Corollary. A space X with a point-countable weak-base is a D-space. Corollary. Any T2 quotient s-image of a metric space is a D-space.