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On the metrizability number and related invariants of spaces. II. (English) Zbl 0864.54001
This paper continues a study begun by these authors in [ibid. 63, 69-77 (1995; Zbl 0860.54005)]. The metrizability number, \(m(X)\), of a space \(X\) is the smallest cardinal \(\kappa\) such that \(X\) can be represented as a union of \(\kappa\) metrizable subspaces. The first countability number, \(fc(X)\), is defined similarly. In this paper a number of conditions on a space \(X\), involving the weight of \(X\) (denoted \(w(X)\)) and \(fc(X)\), are shown to be equivalent to the singular cardinal hypothesis (SCH): if \(2^{\text{cof} (\kappa)}<\kappa\), then \(k^{\text{cof}(\kappa)}=\kappa^+\). For example, SCH is equivalent to the statement “if \(X\) is a space such that \(|X|> 2^\omega\), \(cf(w(X))>\omega\), and \(w(X)\leq |X|\), then \(|X|=w(X) \cdot fc(X)\).” The authors also show that if \(X\) is a compact LOTS and \(m(X)\leq \omega\), then \(X\) is metrizable, and for any compact LOTS \(X\), if \(m(X)>\omega\), then \(w(X) \leq m(X)\). Many other results and examples are given.

54A25 Cardinality properties (cardinal functions and inequalities, discrete subsets)
54D30 Compactness
54A35 Consistency and independence results in general topology
Full Text: DOI
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