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 Free Access

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 and that it is consistent with 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 under the assumption that (whose product with the irrationals is necessarily linearly Lindelöf).

**[A]**K. Alster,*The product of a Lindelöf space with the space of irrationals under Martin’s axiom*, Proc. Amer. Math. Soc.**110**(1990), no. 2, 543–547. MR**993736**, https://doi.org/10.1090/S0002-9939-1990-0993736-9**[AG]**K. Alster and G. Gruenhage,*Products of Lindelöf spaces and GO-spaces*, Topology Appl.**64**(1995), no. 1, 23–36. MR**1339756**, https://doi.org/10.1016/0166-8641(94)00068-E**[BaJ]**Tomek Bartoszyński and Haim Judah,*Set theory*, A K Peters, Ltd., Wellesley, MA, 1995. On the structure of the real line. MR**1350295****[BuMa]**Maxim R. Burke and Menachem Magidor,*Shelah’s 𝑝𝑐𝑓 theory and its applications*, Ann. Pure Appl. Logic**50**(1990), no. 3, 207–254. MR**1086455**, https://doi.org/10.1016/0168-0072(90)90057-9**[K]**Akihiro Kanamori,*The higher infinite*, Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1994. Large cardinals in set theory from their beginnings. MR**1321144****[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), 375–376. MR**0152985**, https://doi.org/10.1090/S0002-9904-1963-10931-3**[M2]**Ernest A. Michael,*Paracompactness and the Lindelöf property in finite and countable Cartesian products*, Compositio Math.**23**(1971), 199–214. MR**0287502****[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]**Sławomir Solecki,*Covering analytic sets by families of closed sets*, J. Symbolic Logic**59**(1994), no. 3, 1022–1031. MR**1295987**, https://doi.org/10.2307/2275926**[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.

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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:
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