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The integers and topology. (English) Zbl 0561.54004
Handbook of set-theoretic topology, 111-167 (1984).
[For the entire collection see Zbl 0546.00022.]
From the introduction: ”We study the role in topology of certain cardinals associated with \(\omega\). We limit our discussion to problems involving first countability, convergence and separable metrizable spaces. Each of our cardinals, m say, is defined to be the minimal cardinality of some special subfamily of P(\(\omega)\) or \(^{\omega}\omega\). The definition is purely set theoretic, and this makes consistency results about m feasible. Our m’s are different from both \(\omega_ 1\) and \({\mathfrak c}\), but only potentially: One has \(\omega_ 1\leq m\leq {\mathfrak c}\) (so \(m=\omega_ 1\) and \(m={\mathfrak c}\) if \({\mathfrak c}=\omega_ 1)\), and each of \(\omega_ 1<m<c\), \(\omega_ 1=m<{\mathfrak c}\), and \(\omega_ 1<m={\mathfrak c}\) is consistent with ZFC... A typical use of these set theoretic cardinals associated with \(\omega\) involves topologically defined cardinals, like the minimum cardinality of a space having a certain property. In the cases we are interested in the topological cardinal, \(\mu\) say, always satisfies \(\omega_ 1\leq \mu \leq {\mathfrak c}\), but consistency results about \(\mu\) seem hard because of \(\mu\) not being set theoretic. However, surprisingly often \(\mu\) is equal to one of those cardinals m associated with \(\omega\), a cardinal which we already know consistency results about, or which is known to be \(\omega_ 1\), in disguise.”
Six main set theoretic (and about forty auxiliary) cardinals are defined and interrelations between them are considered. As the application some estimates of the number of sequentially compact factors is given guaranteed for the sequential compactness or countable compactness of a product. There are constructed a noncompact separable sequentially compact locally compact normal space and a locally compact perfectly normal space, the latter admits a quasi-perfect map onto the rationals that is not inductively perfect, and some others. The article may be recommended to everybody who is interested in set theoretic topology.
Reviewer: M.G.Tkachenko

MSC:
54A25 Cardinality properties (cardinal functions and inequalities, discrete subsets)
54D30 Compactness
54-02 Research exposition (monographs, survey articles) pertaining to general topology
57R65 Surgery and handlebodies
54A35 Consistency and independence results in general topology
54B10 Product spaces in general topology