The Gaps Between the Dimensions of Normal Hausdorff Spaces

Abstract

Let $\mathcal {D}(l, m, n)$ denote the class of all spaces X with ind X = l, $\dim X = m$ and Ind X = n. Recall that (1) $\dim X = 0$ iff Ind X = 0 for any space X, (2) ind X ≤Ind X if X is either T ₁ or regular, and (3) $\dim X \leq {\mathrm {Ind}}\, X$ provided X is a normal space. In this chapter we show that $\mathcal {D}(l, m, n)$ contains a normal Hausdorff space for any triple of integers (l, m, n) that is not ruled out by (1), (2) or (3). We construct a Hausdorff, separable, first countable, ω ₁-compact, countably paracompact and normal member of $\mathcal {D}(l, m, n)$, whenever 0 ≤ l ≤ n and 0 < m ≤ n. The construction will make use of both van Douwen’s technique and Fedorčuk’s method of resolution.