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Paracompactness of metric spaces and the axiom of multiple choice. (English) Zbl 0993.03059
This paper basically considers the statement “metric spaces are paracompact” (MP) and a number (23 to be exact) of variations of MP and related statements. It also considers three versions of the axiom of choice (AC). Its goal is to show which versions of AC are equivalent to which versions of MP. In particular, they focus on MC, the axiom of multiple choice: Given a family of sets \(\mathcal S\), for each \(S \in \mathcal S\) there is finite \(E_S \subset S\). While MC is equivalent to AC in ZF, it is strictly weaker in ZF\(^0\) = ZF with the foundation axiom weakened to allow for ur-elements. The theorem of major interest is: Assume ZF\(^0\). Then MC implies MP, but the reverse implication cannot be proved.

MSC:
03E25 Axiom of choice and related propositions
54D20 Noncompact covering properties (paracompact, Lindelöf, etc.)
54E35 Metric spaces, metrizability
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