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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

The weight of a pseudocompact (homogeneous) space whose cardinality has countable cofinality


Author: Eric K. van Douwen
Journal: Proc. Amer. Math. Soc. 80 (1980), 678-682
MSC: Primary 54A25
DOI: https://doi.org/10.1090/S0002-9939-1980-0587954-5
MathSciNet review: 587954
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Abstract: Let X be an infinite pseudocompact space. We are interested in restrictions on $ \kappa = \vert X\vert$ and $ \lambda = w(X)$ in addition to the obvious inequalities $ \lambda \leqslant {2^\kappa }$ and $ \kappa \leqslant {2^\lambda }$ and $ \kappa \geqslant {2^\omega }$, valid for X without isolated points (in particular for homogeneous X). We show that if $ {\text{cf}}(\kappa ) = \omega $ then $ \lambda \leqslant {2^{ < \kappa }}$, and even $ \lambda \leqslant {2^\mu }$ for some $ \mu < \kappa $ if X is homogeneous. Under the Singular Cardinals Hypothesis (which is much weaker than the GCH), there are no further restrictions for X without isolated points.


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DOI: https://doi.org/10.1090/S0002-9939-1980-0587954-5
Keywords: Weight, cardinality, countably cofinality, strong limit, pseudocompact, homogeneous
Article copyright: © Copyright 1980 American Mathematical Society

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