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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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  • [1] S. H. Hechler, Independence results concerning the number of nowhere dense sets necessary to cover the real line, Acta Math. Acad. Sci. Hungar. 24 (1973), 27-32. MR 47 #1611. MR 0313056 (47:1611)
  • [2] -, Exponents of some $ N$-compact spaces, Israel J. Math. 15 (1973), 384-395. MR 0347594 (50:97)
  • [3] -, On a ubiquitous cardinal, Proc. Amer. Math. Soc. 52 (1975), 348-352. MR 0380705 (52:1602)
  • [4] J. H. B. Kemperman and D. Maharam, $ {R^c}$ is not almost Lindelöf, Proc. Amer. Math. Soc. 24 (1970), 772-773. MR 40 #6500. MR 0253285 (40:6500)
  • [5] D. A. Martin and R. M. Solovay, Internal Cohen extensions, Ann. Math. Logic 2 (1970), 143-178. MR 42 #5787. MR 0270904 (42:5787)
  • [6] R. M. Solovay, A model of set theory in which every set of reals is Lebesgue measurable, Ann of Math. (2) 92 (1970), 1-56. MR 42 #64. MR 0265151 (42:64)

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Additional Information

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

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