Some of the combinatorics related to Michael's problem
Author:
J. Tatch Moore
Journal:
Proc. Amer. Math. Soc. 127 (1999), 24592467
MSC (1991):
Primary 54D20, 54G15
Published electronically:
April 8, 1999
MathSciNet review:
1486743
Fulltext 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 nonLindelöf 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
(90m:54012), http://dx.doi.org/10.1090/S00029939199009937369
 [AG]
K.
Alster and G.
Gruenhage, Products of Lindelöf spaces and GOspaces,
Topology Appl. 64 (1995), no. 1, 23–36. MR 1339756
(96e:54010), http://dx.doi.org/10.1016/01668641(94)00068E
 [BaJ]
Tomek
Bartoszyński and Haim
Judah, Set theory, A K Peters Ltd., Wellesley, MA, 1995. On
the structure of the real line. MR 1350295
(96k:03002)
 [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
(92f:03053), http://dx.doi.org/10.1016/01680072(90)900579
 [K]
Akihiro
Kanamori, The higher infinite, Perspectives in Mathematical
Logic, SpringerVerlag, Berlin, 1994. Large cardinals in set theory from
their beginnings. MR 1321144
(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. 24592467.
 [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
(27 #2956), http://dx.doi.org/10.1090/S000299041963109313
 [M2]
Ernest
A. Michael, Paracompactness and the Lindelöf property in
finite and countable Cartesian products, Compositio Math.
23 (1971), 199–214. MR 0287502
(44 #4706)
 [Mi]
Mi\v{s}\v{c}enko, On Finally Compact Spaces, Soviet Math. Doklady (1962), pp. 24592467.
 [Ru]
M. E. Rudin, Some Conjectures, in: Open Problems In Topology, J. van Mill, G. M. Reed (editors), North Holland (1990), pp. 24592467.
 [S]
Sławomir
Solecki, Covering analytic sets by families of closed sets, J.
Symbolic Logic 59 (1994), no. 3, 1022–1031. MR 1295987
(95g:54033), http://dx.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. 197216.
 [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. 24592467. MR 90m:54012
 [AG]
 K. Alster, G. Gruenhage, Products Of Lindelöf Spaces And GOSpaces, Topology Appl. 64 (1995), pp. 24592467. 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. 24592467. MR 92f:03053
 [K]
 A. Kanamori, The Higher Infinite. Large Cardinals In Set Theory From Their Beginnings, SpringerVerlag (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. 24592467.
 [M1]
 E. Michael, The Product Of A Normal Space And A Metric Space Need Not Be Normal, Bull. Amer. Math. Soc. 69 (1963), pp. 24592467. MR 27:2956
 [M2]
 E. Michael, Paracompactness And The Lindelöf Property In Finite And Countable Cartesian Products, Composito Math. 23 (1971), pp. 24592467. MR 44:4706
 [Mi]
 Mi\v{s}\v{c}enko, On Finally Compact Spaces, Soviet Math. Doklady (1962), pp. 24592467.
 [Ru]
 M. E. Rudin, Some Conjectures, in: Open Problems In Topology, J. van Mill, G. M. Reed (editors), North Holland (1990), pp. 24592467.
 [S]
 S. Solecki, Covering Analytic Sets By Families Of Closed Sets, J. Symbolic Logic, 59 (1994), pp. 24592467. 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. 197216.
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:
http://dx.doi.org/10.1090/S000299399904808X
PII:
S 00029939(99)04808X
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
