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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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[1] Engelking, R., General topology, (1989), Heldermann Verlag Berlin · Zbl 0684.54001
[2] Engelking, R.; Lutzer, D., Paracompactness in ordered spaces, Fund. math., 94, 49-58, (1977) · Zbl 0351.54014
[3] Ismail, M.; Szymanski, A., Compact spaces representable as unions of Nice subspaces, Topology appl., 59, 287-298, (1994) · Zbl 0840.54025
[4] Ismail, M.; Szymanski, A., On the metrizability number and related invariants of spaces, Topology appl., 63, 69-77, (1995) · Zbl 0860.54005
[5] Ismail, M.; Szymanski, A., A topological equivalence of the singular cardinals hypothesis, (), 971-973 · Zbl 0823.54001
[6] Jech, T., Set theory, (1978), Academic Press New York · Zbl 0419.03028
[7] Lutzer, D., On generalized ordered spaces, Dissertationes mathematicae, 89, (1971) · Zbl 0228.54026
[8] Mrowka, S., Some set-theoretic constructions in topology, Fund. math., 94, 83-92, (1977) · Zbl 0348.54017
[9] Todorcevic, S., Trees and linearly ordered sets, (), 235-293 · Zbl 0557.54021
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