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A non-normal box product. (English) Zbl 0328.54017

Infinite finite Sets, Colloq. Honour Paul Erdős, Keszthely 1973, Colloq. Math. Soc. Janos Bolyai 10, 629-631 (1975).
[For the entire collection see Zbl 0293.00009.]
The box product of the family \(\{X_n \mid n \in \omega \}\) 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) \(F= \{f_\alpha \mid \alpha < \kappa \brace\), 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 \(\kappa \neq \omega_1\) 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

MSC:

54D15 Higher separation axioms (completely regular, normal, perfectly or collectionwise normal, etc.)
03E15 Descriptive set theory
54A25 Cardinality properties (cardinal functions and inequalities, discrete subsets)
54G15 Pathological topological spaces

Citations:

Zbl 0293.00009