Another view of metrizability

Abstract

A fact long considered unsatisfactory about the classical metrization theorem of Alexandroff-Urysohn is that it expresses metrizability as a countable uniformity, uniformity itself being almost the former. In view of their unification, the classical theorems, with the exception of Arhangel’skiĭ’s regular open base theorem, are all really subject to the same criticism, to which our theorem here is an answer. We give a generalization here of Arhangel’skiĭ’s, of which Arhangel’skiĭ’s itself, the fundamental theorem of Alexandroff-Urysohn, A. H. Frink’s, and the Double Sequence Theorem of Nagata are all obvious special cases.