Disjoint Unions of Topological Spaces and Choice

Abstract

in the absence of some form of the Axiom of Choice.Mathematics Subject Classification: 03E25, 04A25, 54D10, 54D15.We find properties of topological spaces which are not shared by disjoint unions Introduction and terminologyThis is a continuation of the study of the roll the Axiom of Choice plays in general topology. See also [l], [2], [3], and [4]. 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 Axiomof 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 ')

Paracompactness of Metric Spaces and the Axiom of Multiple Choice

“…(It is the same as the model described in Theorem 10, except that the group G is the group of permutations on A that are rational translations on each Q n .) It is shown in [7] that MM is false in this model. (The argument is similar to the one given in our Theorem 10.)…”

“…( 6) Every open cover U of a metric space can be written as a well ordered union {U α : α ∈ γ} where γ is an ordinal and each U α is point countable. (7) Every metric space has a σ-discrete base. (8) Every metric space has a γ-discrete base, for some ordinal γ.…”

Countable compact Hausdorff spaces need not be metrizable in ZF

“…These statements concern topological sums and Tychonoff products of spaces sharing particular properties and have been studied in [6], [11], and [12] (without the requirement that the coordinate spaces are countable). Proof.…”

The failure of the axiom of choice implies unrest in the theory of Lindelöf metric spaces

“…£ Example 1 demonstrates explicitly the fact that the most fundamental questions concerning Lindelöf (compact) metric spaces can be undecidable from the Zermelo-Frankel axioms of set theory alone. More examples of the counter-intuitive behavior of metric spaces, in absence of AC, can be found in [2,6,3,4,14,12,13,15,8,9,10].…”