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Proceedings of the American Mathematical Society

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

 

 

On $ N\sp{\aleph \sb{1}}$ and the almost-Lindelöf property


Author: Stephen H. Hechler
Journal: Proc. Amer. Math. Soc. 52 (1975), 353-355
MSC: Primary 02K25; Secondary 02K05, 28A35
DOI: https://doi.org/10.1090/S0002-9939-1975-0380706-9
MathSciNet review: 0380706
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Abstract: In 1970, Kemperman and Maharam proved that there exists a Baire measure $ \mu $ on $ {N^{\mathbf{c}}}$ (where $ N$ is the set of natural numbers) such that $ {N^{\mathbf{c}}}$ may be covered by a famliy of elementary open $ \mu $-null sets and used this to prove that $ {R^{\mathbf{c}}}$ (where $ R$ is the set of real numbers) does not have the ``almost-Lindelöf'' property. We define $ {\mathbf{K}}$ to be the smallest cardinal $ \kappa $ for which there exists a collection of $ \kappa $ closed subsets of $ R$ each of Lebesgue measure zero and which covers $ R$, and we show that in the above results $ {\mathbf{c}}$ can be replaced by $ {\mathbf{K}}$. We then note that we have shown elsewhere that it is consistent with the negation of the continuum hypothesis that $ {\mathbf{K}} = {\aleph _1}$, and this, therefore, implies that it is consistent with the negation of the continuum hypothesis that $ {R^\aleph }1$ not be almost-Lindelöf.


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DOI: https://doi.org/10.1090/S0002-9939-1975-0380706-9
Article copyright: © Copyright 1975 American Mathematical Society