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On Stone’s theorem and the Axiom of Choice. (English) Zbl 0893.54016
The context of this highly interesting paper is the following: in Zermelo-Fraenkel set theory, the theorem of Tychonoff, i.e. the statement that any product of compact topological spaces is compact, is equivalent to the axiom of choice. On the other hand, Urysohn’s metrization theorem, that every regular second countable space is metrizable, is provable from just the ZF axioms alone. The present paper considers Stone’s theorem, that every metric space is paracompact, from this perspective by asking how heavily it depends on the axiom of choice. By a forcing argument the authors prove that it is consistent relative to ZF that there exists a (locally connected, locally compact) metric space that is not paracompact. It is even the case that Stone’s theorem cannot be proved from ZF+DC (principle of dependent choice). In the final section the authors consider the question whether ZF+certain weakenings of AC (e.g. Boolean prime ideal theorem, ordering principle, selection principle) imply Stone’s theorem.
Reviewer: M.Ganster (Graz)

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