ORDER-THEORETIC INVARIANTS IN SET-THEORETIC TOPOLOGY

Chapter 1
Introduction
1.1 Van Douwen’s Problem
The original motivation for this entire dissertation was Van Douwen’s Problem, an open
problem in set-theoretic topology.
Definition 1.1.1. A homeomorphism is a continuous bijection with continuous inverse.
Given a topological space X, let Aut(X) denote the group of autohomeomorphisms of
X. A space X is homogeneous if for every p, q ∈ X, there exists h ∈ Aut(X) such that
h(p) = q.
Definition 1.1.2. A compactum is a compact Hausdorff space.
Question 1.1.3 (Van Douwen’s Problem). Is there a homogeneous compactum X and a
family F of pairwise disjoint open subsets of X such that F has greater cardinality than
R?
This problem has been open (in all models of ZFC) for over thirty years [47]. To get
an idea of why problems about homogeneous compacta can be so hard, ask, given an
arbitrary list 〈Xi〉i∈I of homogeneous compacta, what can we do with them to produce a
bigger homogeneous compacta? In general, all we know how to do is form products like
i∈I Xi. (Actually, Chapter 2 describes a method for producing homogeneous quotients
1