On the metrizability number and related invariants of spaces. II (Q1924681)

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.