Failure of the Subset Theorem for Hereditarily Normal Spaces

Abstract

The subset theorem for the covering dimension of a space X is the statement that \(\dim A \leq \dim X\) for every subspace A of X. The subset theorem for perfectly normal spaces was established by Cech. The same author in 1948 raised the question whether the subset theorem holds for hereditarily normal Hausdorff spaces (see Cech, Colloq Math 1:332, 1948). Recall that there are trivial examples of hereditarily normal (but neither T1 nor regular) spaces for which the subset theorem fails (Exercise 2.18). Also, for any \(n\in \mathbb {N}\), there is a strongly zero-dimensional normal Hausdorff space N that contains a normal space M with \(\dim M = n\) (Theorem 14.9). A negative answer to Cech’s question was first obtained by Filippov in 1973, who assumed, however, the existence of a Souslin tree. The problem was finally settled in ZFC, without additional set theoretic assumptions, by Elzbieta and Roman Pol in 1977. This chapter is devoted to presenting their example of a Hausdorff, strongly zero-dimensional, hereditarily normal space that contains a perfectly normal, locally second countable subspace of positive covering dimension.