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

DOI:
https://doi.org/10.1090/S0002-9939-1979-0521889-0

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. or ), then it is known that even perfectly normal such examples exist.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1979-0521889-0

Keywords:
First countable,
normal,
ccc,
separable

Article copyright:
© Copyright 1979
American Mathematical Society