Abstract
This chapter discusses transfinite dimension. Applying transfinite induction, it can be easily verify that the small transfinite dimension is a topological invariant. The chapter describes three classes of spaces that are only slightly larger than the class of finite-dimensional spaces. It also discusses basic properties of transfinite dimensions. The chapter highlights metric spaces with large transfinite dimension. The class of separable metric spaces that have small transfinite dimension coincides with the class of spaces which have countably dimensional metrizable compactifications, but this is not an internal characterization. The chapter also discusses the relations between the existence of small and large transfinite dimensions in separable metric spaces.
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