Urysohn's metrization theorem for higher cardinals

Abstract

In this paper a generalization of Urysohn's metrization theorem is given for higher cardinals. Namely, it is shown that a topological space with a basis of cardinality at most $|{\omega }_{\mu }|$ or smaller is ${\omega }_{\mu }$-metrizable if and only if it is ${\omega }_{\mu }$-additive and regular, or, equivalently, ${\omega }_{\mu }$-additive, zero-dimensional, and T\textsubscript{0}. Furthermore, all such spaces are shown to be embeddable in a suitable generalization of Hilbert's cube.

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