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A ZFC Dowker space in \(\aleph_{\omega+1}\): An application of pcf theory to topology. (English) Zbl 0895.54022

As is well known, a Dowker space is one that is normal but whose product with the unit interval is not normal. Dowker spaces exist; there is one of size \(\aleph_\omega^{\aleph_0}\), due to M. E. Rudin [Fundam. Math. 73, 179-186 (1971; Zbl 0224.54019)] and two of size \(2^{\aleph_0}\), due to Z. T. Balogh [Proc. Am. Math. Soc. 124, No. 8, 2555-2560 (1996; Zbl 0876.54016); A normal screenable nonparacompact space in ZFC, ibid. 126, 1835-1844 (1998)]. Before and after Rudin’s example was published there has been an ongoing search for ‘small’ Dowker spaces, e.g., separable or of size \(\aleph_1\). A comprehensive list of such examples constructed from Souslin trees, \(\clubsuit\)-sequences etc. can be found in [P. J. Szeptycki and W. A. R. Weiss, Ann. N. Y. Acad. Sci. 705, 119-129 (1993; Zbl 0847.54022)].
This paper presents the first ZFC example of a ‘truly small’ Dowker space: it is of well-determined cardinality \(\aleph_{\omega+1}\) – in contrast with the earlier examples, whose cardinality may be weakly inaccessible. The starting point is the second author’s theorem that there is a subset \(B\) of \(\omega\) such that in \(\prod_{n\in B}\aleph_n\) there is a sequence \(\langle f_\alpha:\alpha<\aleph_{\omega+1}\rangle\) that is increasing and cofinal with respect to the mod finite order \(<^*\). This then is turned into a closed cofinal subspace \(X\) of Rudin’s example \(R\); the latter is a subspace of \(\prod_{n\in B}\aleph_n\) with the box topology. Because \(X\) is closed in \(R\) it is normal and because \(X\) is cofinal in \(R\) its product with \([0,1]\) is not normal. In spite of some strange misprints well worth reading; the example is nice and it makes one curious about pcf theory.
Reviewer: K.P.Hart (Delft)

MSC:

54G20 Counterexamples in general topology
54B99 Basic constructions in general topology
03E05 Other combinatorial set theory
03E50 Continuum hypothesis and Martin’s axiom
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