Topologies on omega-1 and guessing sequences

Abstract

We show the solution of an over 25 years old conjecture of Blair that MA + ¬CH implies that every perfectly normal space of cardinality less than the first measurable cardinal (if it exists) is realcompact. To solve this problem we construct a topology on w1 associated with a guessing sequence which is a sequence ⟨E _α : α ∈ Lim (ω₁)⟩ of cofinal subsets of α, E_α, for each α ∈ Lim (ω₁). We present a systematic study of the properties of these topological spaces and the combinatorial properties of the guessing sequences that we use to define them. We give the construction of several guessing sequences using Gödel's Constructibility Axiom and by the forcing method. Besides the construction of a perfectly normal space of cardinality [special characters omitted] that is not realcompact and is compatible with MA + ¬CH, we also present the construction of a new Dowker space. It has also been a long outstanding problem in the theory of Dowker spaces whether MA + ¬CH, or even PFA, implies that there is no Dowker space of cardinality [special characters omitted]. While we have not succeeded solving this problem we hope that our method of constructing this new Dowker space will lead to an example of a Dowker space of cardinality [special characters omitted] compatible with MA + ¬CH. Likewise, with the purpose to show the flexibility of our technique we also construct a new anti-Dowker space as well as other topological spaces. We hope that new applications of our methods will be found shortly.