A Normal First Countable ccc Nonseparable Space

Abstract

We construct an absolute example of a space having the properties in the title. Let Y be the set of nonempty finite subsets of the Cantor cube of countable weight. The Pixley-Roy topology on Y is not normal, but the Vietoris topology on Y is normal. Our space can be considered a normalization of the Pixley-Roy topology on Y by adding cluster points which as a subspace have the Vietoris topology. The Alexandroff duplicating procedure is used liberally to glue the space together. The example is also a sigma compact paracompact p-space. If further set-theoretic assumptions are made (e.g. $V = L$ or ${\text {MA}} + \neg {\text {CH}}$), then it is known that even perfectly normal such examples exist.