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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

A separable space with no remote points


Author: Alan Dow
Journal: Trans. Amer. Math. Soc. 312 (1989), 335-353
MSC: Primary 54D35; Secondary 03E35, 03E55, 54A35, 54D40, 54D60
DOI: https://doi.org/10.1090/S0002-9947-1989-0983872-1
MathSciNet review: 983872
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Abstract: In the model obtained by adding $ {\omega _2}$ side-by-side Sacks reals to a model of $ {\mathbf{CH}}$, there is a separable nonpseudocompact space with no remote points. To prove this it is also shown that in this model the countable box product of Cantor sets contains a subspace of size $ {\omega _2}$ such that every uncountable subset has density $ {\omega _1}$. Furthermore assuming the existence of a measurable cardinal $ \kappa $ with $ {2^\kappa } = {\kappa ^ + }$, a space $ X$ is produced with no isolated points but with remote points in $ \upsilon X - X$. It is also shown that a pseudocompact space does not have remote points.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1989-0983872-1
Keywords: Remote points, measurable cardinals, side-by-side Sacks forcing
Article copyright: © Copyright 1989 American Mathematical Society