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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Some of the combinatorics
related to Michael's problem


Author: J. Tatch Moore
Journal: Proc. Amer. Math. Soc. 127 (1999), 2459-2467
MSC (1991): Primary 54D20, 54G15
Published electronically: April 8, 1999
MathSciNet review: 1486743
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Abstract: We present some new methods for constructing a Michael space, a regular Lindelöf space which has a non-Lindelöf product with the space of irrationals. The central result is a combinatorial statement about the irrationals which is a necessary and sufficient condition for the existence of a certain class of Michael spaces. We also show that there are Michael spaces assuming $\mathfrak{d}= \operatorname{cov}(\mathcal{M})$ and that it is consistent with $\operatorname{cov}(\mathcal{M}) < \mathfrak{b} < \mathfrak{d}$ that there is a Michael space. The influence of Cohen reals on Michael's problem is discussed as well. Finally, we present an example of a Michael space of weight less than $\mathfrak{b}$ under the assumption that $\mathfrak{b} = \mathfrak{d}= \operatorname{cov} (\mathcal{M}) = \aleph _{\omega +1}$ (whose product with the irrationals is necessarily linearly Lindelöf).


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Additional Information

J. Tatch Moore
Affiliation: Department of Mathematics, University of Toronto, Toronto, Canada M5S 1A1
Email: justin@math.toronto.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-99-04808-X
PII: S 0002-9939(99)04808-X
Keywords: Lindel\"of, linearly Lindel\"of, irrationals, Michael space.
Received by editor(s): September 22, 1997
Received by editor(s) in revised form: October 29, 1997
Published electronically: April 8, 1999
Communicated by: Alan Dow
Article copyright: © Copyright 1999 American Mathematical Society