Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9939-99-04808-X
Published electronically: April 8, 1999
MathSciNet review: 1486743
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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).


References [Enhancements On Off] (What's this?)

  • [A] K. Alster, The Product Of A Lindelöf Space With The Space Of Irrationals Under Martin's Axiom, Proc. Amer. Math. Soc. 110 (1990), pp. 2459-2467. MR 90m:54012
  • [AG] K. Alster, G. Gruenhage, Products Of Lindelöf Spaces And GO-Spaces, Topology Appl. 64 (1995), pp. 2459-2467. MR 96e:54010
  • [BaJ] T. Bartoszy\'{n}ski, H. Judah, Set Theory: On The Structure Of The Real Line, A. K. Peters (1995). MR 96k:03002
  • [BuMa] M. Burke, M. Magidor, Shelah's pcf Theory And It's Applications, Ann. Pure. Appl. Logic 50 (1990), pp. 2459-2467. MR 92f:03053
  • [K] A. Kanamori, The Higher Infinite. Large Cardinals In Set Theory From Their Beginnings, Springer-Verlag (1994). MR 96k:03125
  • [L] B. Lawrence, The Influence Of A Small Cardinal On The Product Of A Lindelöf Space And The Irrationals, Proc. Amer. Math. Soc. 110 (1990), pp. 2459-2467.
  • [M1] E. Michael, The Product Of A Normal Space And A Metric Space Need Not Be Normal, Bull. Amer. Math. Soc. 69 (1963), pp. 2459-2467. MR 27:2956
  • [M2] E. Michael, Paracompactness And The Lindelöf Property In Finite And Countable Cartesian Products, Composito Math. 23 (1971), pp. 2459-2467. MR 44:4706
  • [Mi] Mi\v{s}\v{c}enko, On Finally Compact Spaces, Soviet Math. Doklady (1962), pp. 2459-2467.
  • [Ru] M. E. Rudin, Some Conjectures, in: Open Problems In Topology, J. van Mill, G. M. Reed (editors), North Holland (1990), pp. 2459-2467.
  • [S] S. Solecki, Covering Analytic Sets By Families Of Closed Sets, J. Symbolic Logic, 59 (1994), pp. 2459-2467. MR 95g:54033
  • [V] J. Vaughan, Small Uncountable Cardinals And Topology, in: Open Problems In Topology, J. van Mill, G. M. Reed (editors), North Holland (1990) pp. 197-216.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 54D20, 54G15

Retrieve articles in all journals with MSC (1991): 54D20, 54G15


Additional Information

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

DOI: https://doi.org/10.1090/S0002-9939-99-04808-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

American Mathematical Society