The Inductive Dimension Ind ₀

Abstract

Replacing open covers with cozero covers in the definition of $\dim$ resulted in the dimension function dim₀ studied in Chap. 11. Replacing partitions with zero partitions in the definition of inductive dimensions results in the two new inductive dimension functions, Ind₀ and Ind₀, studied in this chapter. We show that Ind₀ satisfies the countable sum theorem for zero subsets of a normal space and the subset theorem for a normal, z-embedded subset of any space. The countable sum theorem and the subset theorem for $\mathop {\mathrm {Ind}} \nolimits$ on perfectly normal spaces are immediate corollaries. For a regular, Lindelöf space X, we show that Ind₀ X = Ind₀ X. On several occasions the equality Ind₀ X = Ind₀ X in conjunction with the sum theorem and other properties of Ind₀ will prove useful in estimating both Ind₀ and $\mathop {\mathrm {Ind}} \nolimits$. After establishing the theoretical results, we present an example of a hereditarily normal, compact Hausdorff space T _n with $\mathop {\mathrm {Ind}} \nolimits T_n = 1$ and Ind₀ T _n = Ind₀ T _n = n, for each $n\in \mathbb {N}$. Moreover, T _n is the union of 2ⁿ⁻¹ closed subspaces with $\mathop {\mathrm {Ind}} \nolimits _0= 1$. Note that $\mathop {\mathrm {Ind}} \nolimits$, too, fails to satisfy the finite sum theorem for closed subsets: In Chap. 14, there is a compact Hausdorff space S with $\mathop {\mathrm {Ind}} \nolimits S= 2$ which is the union of two closed subsets with $\mathop {\mathrm {Ind}} \nolimits = 1$.