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**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 **{**X_{n} | n in \omega **}** of topological spaces is just **prod**_{n in \omega} X_{n} with a basis consisting of arbitrary products of open sets. The paper concerns families where each X_{n} 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 \omega_{1} is the minimal cardinality of a scale then **prod** X_{n} is not normal where X_{0} = \kappa and X_{n} = \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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