Subsets of \(^{\omega}\omega\) and generalized metric spaces
Recall the Fréchet order on \(^{\omega}\omega:f<^*g\) iff, for all but finitely many n, \(f(n)<g(n)\). A \(\kappa\)-scale is a well-ordered dominating (under the Fréchet order) family of type \(\kappa\). The statement BF(\(\kappa)\) says that if \(A\subset^{\omega}\omega\) and \(| A| <\kappa\) then there is some g with \(f<^*g\) for all \(f\in A\). CH implies both ''there is an \(\omega_ 1\)-scale'' and BF(\({\mathfrak c})\), where \({\mathfrak c}\) is the cardinality of the continuum. In this paper, the authors derive statements about generalized metric spaces known to follow from CH from statements about the Fréchet order. For example: using Heath's bow-tie space, they get a Lindelöf semimetrizable space with no countable network from an \(\omega_ 1\)- scale; from BF(\(\kappa)\) they get ''every sequential space of character less than \(\kappa\) is Fréchet''; and from BF(\({\mathfrak c})\) they get ''every compact Hausdorff space which is weakly first countable has size either \(\leq \omega\) or \(\geq {\mathfrak c}.''\)
scientific article; zbMATH DE number 3884147