ψ-Spaces and the Failure of the Sum and Subset Theorems for dim ₀

Abstract

The countable sum theorem for $\dim$ is formulated in terms of closed subsets of a normal space while in a corresponding result for dim₀ closed sets are replaced with z-embedded zero subspaces. The question arises whether it is necessary for the zero subspaces to be z-embedded. In this chapter, for any given $n \in \mathbb {N}$, we present a Tychonoff space X which is the union of two zero subspaces X ₁, X ₂ such that dim₀ X ₁ = dim₀ X ₂ = 0 while dim₀ X = n. We also construct Tychonoff spaces Y  with dim₀ Y = 0 that contain zero subspaces Z with dim₀ Z as large as we wish, showing the failure of the subset theorem for dim₀ in a strong form.