Paracompactness of Metric Spaces and the Axiom of Multiple Choice

The axiom of multiple choice implies that metric spaces are paracompact but the reverse implication cannot be proved in set theory without the axiom of choice. 1. Background, Definitions and Summary of Results. Working in set theory without the axiom of choice we study the deductive strength of the assertion MP: Metric spaces are paracompact. (Definitions are given below.) MP was first proved in 1948 by A. H. Stone ([17]) using the axiom of choice (AC). A considerably shortened proof was given by Mary Ellen Rudin in [15]. In Rudin’s proof the use of the axiom of choice is evident since the proof uses a wellordering of an arbitrary open cover of a metric space. More recently, Good and Tree have shown that a metric space is paracompact provided it has a well-ordered dense subset. If we let (∗) and (∗∗) represent the following statements: (∗) Every open cover of every metric space can be well-ordered. (∗∗) Every metric space has a well-ordered dense subset. then the results of Rudin and Good/Tree can be written respectively as (∗) ⇒ MP and (∗∗) ⇒ MP. Both are theorems in set theory without the axiom of choice. However (∗) and (∗∗) are both equivalent to the axiom of choice. (Let X be any set and d the metric defined on X by d(y, z) = 1 if y 6= z. Since X is the only dense subset of X , (∗∗) gives a well-ordering of X . Similarly, applying (∗) to the open cover {{x} : x ∈ X} gives a well ordering of X .) Therefore neither the theorem of Rudin nor that of Good and Tree help us in placing MP in the deductive hierarchy of weak versions of the axiom of choice. (Although, as we will mention later, several of our results rely heavily on the proof given in [15].) Some progress has been made in determining the deductive strength of MP. Recently Good, Tree and Watson [5] have constructed models of both ZermeloFraenkel set theory (ZF) and Zermelo-Fraenkel set theory with the axiom of foundation modified to permit the existence of atoms (ZF) in which MP is false. One of our purposes in this paper is to show that the axiom of multiple choice (MC) implies MP. This is one of our theorems that depends on [15]. We will also 1991 Mathematics Subject Classification. 03E25, 04A25, 54D10, 54D15.