ℵ‐completeness as E‐compactness via the Uniform Dimensiona

Abstract

Let ℵ be an infinite cardinal number and ℛ (ℵ, 1) a complete metric space of density ℵ and uniform dimension 1 uniformly universal in the class of all metric spaces of density ≤ℵ and uniform dimension ≤ 1. The ℵ‐uniformity of a Tychonoff space X, which is generated by all open normal coverings of X with cardinality ≤ℵ, is complete iff X can be embedded as a closed subspace in a Cartesian product of copies of ℛ (ℵ, 1).