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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.

##### MSC:
 54A25 Cardinality properties (cardinal functions and inequalities, discrete subsets) 54D30 Compactness 54A35 Consistency and independence results in general topology
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##### References:
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