No Compactification Theorem for the Small Inductive Dimension of Perfectly Normal Spaces

Abstract

Recall that for a normal Hausdorff space X, we have $\dim X = \dim \beta X$ and $\mathop {\mathrm {Ind}} \nolimits X = \mathop {\mathrm {Ind}} \nolimits \beta X$. Moreover, a T ₁ zero-dimensional space X can be embedded in a Cantor cube and so X has a zero-dimensional compactification. It is therefore natural to ask whether every normal Hausdorff space X has a compactification Y  with $\mathop {\mathrm {ind}} \nolimits Y = \mathop {\mathrm {ind}} \nolimits X$. In this chapter we construct a Hausdorff, perfectly normal space X with $\mathop {\mathrm {ind}} \nolimits X= 1$ such that $\dim Y = \mathop {\mathrm {ind}} \nolimits Y= \infty$ for every compactification Y  of X. The first such example is due to van Mill and Przymusiński (Topol Appl 13:133–136, 1982). Bear in mind that by Proposition 5.3, $\dim Y \leq \mathop {\mathrm {ind}} \nolimits Y$ for every Lindelöf space Y .