FIRST COUNTABLE SPACES WITH CALIBER ℵ1 MAY OR MAY NOT BE SEPARABLE

Abstract

This chapter discusses the first countable vspaces with caliber 81. A space has caliber 81 if every uncountable collection of open sets includes an uncountable subcollection with nonempty intersection. This property lies strictly between the countable chain condition and separability. It is proved that the continuum hypothesis implies first countable Hausdorff spaces with caliber 81 are separable. A simple proof can be obtained by noting that first countable Hausdorff spaces satisfying the countable chain condition have cardinality ≤280 [J, 2.16] and that spaces of cardinality 81 with caliber 81 are separable.