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Abstract

We construct a family of spaces with "nice" structure which is universal in the class of all compact metrizable spaces of large transfinite dimension $ω_0$, or, equivalently, of small transfinite dimension $ω_0$; that is, the family consists of compact metrizable spaces whose transfinite dimension is $ω_0$, and every compact metrizable space with transfinite dimension $ω_0$ is embeddable in a space of the family. We show that the least possible cardinality of such a universal family is equal to the least possible cardinality of a dominating sequence of irrational numbers.