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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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On $N^{\aleph _{1}}$ and the almost-Lindelöf property
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by Stephen H. Hechler PDF
Proc. Amer. Math. Soc. 52 (1975), 353-355 Request permission

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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Additional Information
  • © Copyright 1975 American Mathematical Society
  • 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