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For every cardinal τ and every ordinal α, we construct a metrizable space $M_{\alpha }\left(\tau \right)$ and a strongly countable-dimensional compact space $Z_{\alpha }\left(\tau \right)$ of weight τ such that $D\left(M_{\alpha }\left(\tau \right)\right)\le \alpha$, $D\left(Z_{\alpha }\left(\tau \right)\right)\le \alpha$ and each metrizable space X of weight τ such that D(X) ≤ α is homeomorphic to a subspace of $M_{\alpha }\left(\tau \right)$ and to a subspace of $Z_{\alpha +1}\left(\tau \right)$.

We construct a family of spaces with “nice” structure which is universal in the class of all compact metrizable spaces of large transfinite dimension ${\omega }_0$, or, equivalently, of small transfinite dimension ${\omega }_0$; that is, the family consists of compact metrizable spaces whose transfinite dimension is ${\omega }_0$, and every compact metrizable space with transfinite dimension ${\omega }_0$ 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...

R. Pol has shown that for every countable ordinal α, there exists a universal space for separable metrizable spaces X with ind X = α . We prove that for every countable limit ordinal λ, there is no universal space for separable metrizable spaces X with Ind X = λ. This implies that there is no universal space for compact metrizable spaces X with Ind X = λ. We also prove that there is no universal space for compact metrizable spaces X with ind X = λ.

For every countable non-limit ordinal α we construct an α-dimensional Cantor ind-manifold, i.e., a compact metrizable space $Z_{\alpha }$ such that $ind{Z}_{\alpha }=\alpha$, and no closed subset L of $Z_{\alpha }$ with ind L less than the predecessor of α is a partition in $Z_{\alpha }$. An α-dimensional Cantor Ind-manifold can be constructed similarly.