Urysohns metrization theorem for higher cardinals

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.

💡 Research Summary

The paper presents a comprehensive generalization of the classical Urysohn metrization theorem to higher cardinalities. In the classical setting, a topological space that possesses a countable base and is regular (or equivalently, zero‑dimensional, $T_0$, and has a countable base) is metrizable. The author replaces “countable” with an arbitrary infinite cardinal $\omega_\mu$ (where $\mu$ is an ordinal) and develops the corresponding notions of $\omega_\mu$‑additivity, $\omega_\mu$‑metrizable spaces, and a generalized Hilbert cube $