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# A Normal First Countable ccc Nonseparable Space

Murray G. Bell
Proceedings of the American Mathematical Society
Vol. 74, No. 1 (Apr., 1979), pp. 151-155
DOI: 10.2307/2042121
Stable URL: http://www.jstor.org/stable/2042121
Page Count: 5
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 $\mathrm{MA} + \neg \mathrm{CH}$), then it is known that even perfectly normal such examples exist.