G-3 - Dowker Spaces

Abstract

A normal Hausdorff space X is said to be a Dowker space if X × [0, 1] is not normal. The problem of the existence of such a space originally arose from homotopy extension considerations, but soon became a difficult problem that propelled the development of set-theoretic topology. It is easy to prove that a normal space X is a Dowker space if and only if (iff) it has an increasing open cover (Un)n∈ω such that there is no closed cover (Fn) n∈ω of X with Fn ⊂Un for every n ∈ ω. This chapter discusses the construction of Dowker spaces in Zermelo–Fraenkel set theory with the axiom of choice (ZFC) and describes two basic methods of construction in detail. Some applications are also mentioned in the chapter. In spite of the wide variety and long history of these constructions, there has been, in essence, only one method used for the construction of Dowker spaces in special models of ZFC.