Chapter 59 - Some problems in the dimension theory of compacta

Abstract

This chapter discusses some problems in the dimension theory of compacta. All topological spaces considered in this chapter are assumed to be Tychonoff and called simply spaces; maps mean continuous maps of topological spaces. Almost all problems posed in the chapter concern compact spaces. The chapter defines the dimension Δ of a paracompact space X as: ΔX ≤n if there exists a strongly zero-dimensional paracompact space X0 and a surjective closed map f : X0 →X such that |f−1x| ≤n +1 for any x ∈X. It is known that the three basic dimension functions dim, ind, and Ind coincide for compact metrizable spaces, that is, dim X = ind X = Ind X for any compact metrizable space X. In 1936, Alexandroff asked whether they coincide for arbitrary compact spaces. In 1941, he proved that dim X ≤ ind X for any compact space X. Also Ind X ≤ Ind X for any normal space X and Ind X ≤ ΔX for any paracompact space X. The chapter discusses about on the coincidence of dim, ind, Ind, and Δ for compact spaces. It explains noncoincidence of dim and ind for compact spaces as well as noncoincidence of ind and Ind for compact spaces. A detailed discussion on dimensional properties of topological products is also presented.