Every Separable Metrizable Space has a Proper Dyadic Subbase

Abstract

The notions of a proper dyadic subbase and an independent subbase was introduced by H. Tsuiki to investigate in {0, 1, bot}-sequence codings of topological spaces. We show that every separable metrizable space has a proper dyadic subbase whose restriction to the perfect set defined by the Cantor-Bendixson theorem forms an independent subbase of the restricted space.