Zentralblatt MATH

Publications of (and about) Paul Erdös

Zbl.No:  328.54017
Autor:  Erdös, Paul; Rudin, Mary Ellen
Title:  A non-normal box product. (In English)
Source:  Infinite finite Sets, Colloq. Honour Paul Erdös, Keszthely 1973, Colloq. Math. Soc. Janos Bolyai 10, 629-631 (1975).
Review:  [For the entire collection see Zbl 293.00009.]
The box product of the family {Xn | n in \omega } of topological spaces is just prodn in \omega Xn with a basis consisting of arbitrary products of open sets. The paper concerns families where each Xn is an ordinal with the order topology. A subset F of \omega \omega is a \kappa-scale if i) F = {f\alpha | \alpha < \kappa \brace, ii) \alpha < \beta < \kappa implies f\alpha (n) < f\beta (n) for all but finitely many n and iii) for any f in \omega \omega there are an \alpha < \kappa and an m < \omega with f(m) < f\alpha (m) \forall m > n. The following is proven. Theorem. If \kappa \ne \omega1 is the minimal cardinality of a scale then prod Xn is not normal where X0 = \kappa and Xn = \omega+1 all n > 0. The paper also states a number of facts about such spaces.
Reviewer:  J.M.Plotkin
Classif.:  * 54D15 Higher separation axioms
                   03E15 Descriptive set theory (logic)
                   04A15 Descriptive set theory
                   54A25 Cardinality properties of topological spaces
                   54G15 Pathological spaces

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