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A normal first countable ccc nonseparable space

Author: Murray G. Bell
Journal: Proc. Amer. Math. Soc. 74 (1979), 151-155
MSC: Primary 54D15; Secondary 54G20
MathSciNet review: 521889
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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.

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Keywords: First countable, normal, ccc, separable
Article copyright: © Copyright 1979 American Mathematical Society