A Regular Countable Chain Condition Space without Compact‐caliber (ω1, ω)

Abstract

ABSTRACT. A topological space has the countable chain condition (CCC) if every disjoint collection of nonempty open sets is countable. It has compact‐caliber (ω1, ω) if, for every family {Uα: α∈ω1) of nonempty open sets, there is a compact set K such that K ∩ Uα |Mn Ø for infinitely many α∈ω1. It has been previously shown that CCC implies compact‐caliber (ω1, ω) for first countable regular spaces. An example is constructed to show that CCC does not imply compact‐caliber (ω1, ω) for arbitrary regular spaces. The method of construction is to refine the usual topology on the set of real numbers, and take the Pixley‐Roy space over this refinement.