Urysohn's metrization theorem for higher cardinals

TL;DR

This paper extends Urysohn's metrization theorem to higher cardinal spaces, characterizing $oldsymbol{ ext{omega}_oldsymbol{ u}}$-metrizable spaces via additive and regular properties, and embedding them into generalized Hilbert cubes.

Contribution

It provides a new characterization of higher cardinal metrizable spaces and demonstrates their embeddability into generalized Hilbert cubes, extending classical metrization results.

Findings

Spaces with basis size ≤ | extomega_ u| are $ extomega_ u$-metrizable if and only if they are $ extomega_ u$-additive and regular.

Such spaces are also characterized as $ extomega_ u$-additive, zero-dimensional, T0 spaces.

All these spaces can be embedded into a generalized Hilbert cube.

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.