Disjoint Unions of Topological Spaces and Choice

We find properties of topological spaces which are not shared by disjoint unions in the absence of some form of the Axiom of Choice. Introduction and Terminology This is a continuation of the study of the roll the Axiom of Choice plays in general topology. See also [vd], [gt], [wgt], and [hkrr]. Our primary concern will be the use of the axiom of choice in proving properties of disjoint unions of topological spaces (See Definition 1, part 11.) For example, in set theory with choice the disjoint union of metrizable topological spaces is a metrizable topological space. The usual proof of this fact begins with the choice of metrics for the component spaces. We will show that the use of some form of choice cannot be avoided in this proof and in fact without choice the disjoint union of metrizable spaces may not even be metacompact. In section 1 we show that many assertions about disjoint unions of topological spaces are equivalent to the axiom of multiple choice. Models of set theory and corresponding independence results are described in section 2. In section 3, we study the roll the Axiom of Choice plays in the properties of disjoint unions of collectionwise Hausdorff and collectionwise normal spaces. We begin with the definitions of the symbols and terms we will be using.