A Lindelöf space with no Lindelöf subspace of size
Authors:
Piotr Koszmider and Franklin D. Tall
Journal:
Proc. Amer. Math. Soc. 130 (2002), 27772787
MSC (2000):
Primary 54A20, 54A25, 54A35; Secondary 03E35
Published electronically:
March 13, 2002
MathSciNet review:
1900885
Fulltext PDF Free Access
Abstract 
References 
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Additional Information
Abstract: A consistent example of an uncountable Lindelöf (and hence normal) space with no Lindelöf subspace of size is constructed. It remains unsolved whether extra settheoretic assumptions are necessary for the existence of such a space. However, our space has size and is a space, i.e., 's are open, and for such spaces extra settheoretic assumptions turn out to be necessary.
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J. E. Baumgartner, F. D. Tall; Reflecting Lindelöfness; Top. Appl. In press.
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E. Van Douwen; The Integers and Topology; in [KV] pp. 111  167. MR 87f:54008
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R. Engelking; General Topology; Heldermann Verlag, Berlin 1989. MR 91c:54001
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I. Gorelic; The Baire category and forcing large Lindelöf spaces with points ; Proc. Amer. Math. Soc. 118, 1993, No 2. pp. 603607. MR 93g:03046
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A. Hajnal, I. Juhász; Remarks on the cardinality of compact spaces and their Lindelöf subspaces; Proc. Amer. Math. Soc. 59, 1976, pp. 146  148. MR 54:11263
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I. Juhász, W. Weiss; On a problem of Sikorski; Fund. Math. 100, 1978, pp. 223  227. MR 80g:54006
 [JuW]
W. Just, M. Weese; Discovering Modern SetTheory. Settheoretic tools for every mathematician. Vol II. Graduate Studies in Mathematics. Vol 18. American Mathematical Society, Providence, RI, 1997. MR 99b:03001
 [K]
K.Kunen; Set Theory; An Introduction to Independence Proofs; North Holland, 1983. MR 85e:03003
 [KV]
K. Kunen, J. Vaughan eds. Handbook of settheoretic topology; North Holland, 1984. MR 85k:54001
 [R]
M. E. Rudin; Lectures on SetTheoretic Topology; American Mathematical Society, Providence, 1975. MR 51:4128
 [S]
S. Shelah; On some problems in general topology pp. 91  101, in Set Theory; Boise, Idaho; 1992  1994; American Mathematical Society; Providence 1996. MR 96k:03120
 [T]
S.Todorcevic; Trees, subtrees and order types; Ann. Math. Logic 20, 1981, pp. 233268. MR 82m:03062
 [W]
W. Weiss; Versions of Martin's Axiom; in [KV] pp. 827  886. MR 86h:03088
 [A]
 A. Arhangel'ski; On the cardinality of bicompacta satisfying the first axiom of countability; Soviet Math. Dokl. 10, 1969, pp. 951  955.
 [BT]
 J. E. Baumgartner, F. D. Tall; Reflecting Lindelöfness; Top. Appl. In press.
 [vD]
 E. Van Douwen; The Integers and Topology; in [KV] pp. 111  167. MR 87f:54008
 [E]
 R. Engelking; General Topology; Heldermann Verlag, Berlin 1989. MR 91c:54001
 [G]
 I. Gorelic; The Baire category and forcing large Lindelöf spaces with points ; Proc. Amer. Math. Soc. 118, 1993, No 2. pp. 603607. MR 93g:03046
 [HJ]
 A. Hajnal, I. Juhász; Remarks on the cardinality of compact spaces and their Lindelöf subspaces; Proc. Amer. Math. Soc. 59, 1976, pp. 146  148. MR 54:11263
 [JW]
 I. Juhász, W. Weiss; On a problem of Sikorski; Fund. Math. 100, 1978, pp. 223  227. MR 80g:54006
 [JuW]
 W. Just, M. Weese; Discovering Modern SetTheory. Settheoretic tools for every mathematician. Vol II. Graduate Studies in Mathematics. Vol 18. American Mathematical Society, Providence, RI, 1997. MR 99b:03001
 [K]
 K.Kunen; Set Theory; An Introduction to Independence Proofs; North Holland, 1983. MR 85e:03003
 [KV]
 K. Kunen, J. Vaughan eds. Handbook of settheoretic topology; North Holland, 1984. MR 85k:54001
 [R]
 M. E. Rudin; Lectures on SetTheoretic Topology; American Mathematical Society, Providence, 1975. MR 51:4128
 [S]
 S. Shelah; On some problems in general topology pp. 91  101, in Set Theory; Boise, Idaho; 1992  1994; American Mathematical Society; Providence 1996. MR 96k:03120
 [T]
 S.Todorcevic; Trees, subtrees and order types; Ann. Math. Logic 20, 1981, pp. 233268. MR 82m:03062
 [W]
 W. Weiss; Versions of Martin's Axiom; in [KV] pp. 827  886. MR 86h:03088
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Additional Information
Piotr Koszmider
Affiliation:
Departamento de Matemática, Universidade de São Paulo, Caixa Postal: 66281, São Paulo, SP, CEP: 05315970, Brasil
Email:
piotr@ime.usp.br
Franklin D. Tall
Affiliation:
Department of Mathematics, University of Toronto, Toronto, Ontario, Canada M5S 3G3
Email:
tall@math.toronto.edu
DOI:
http://dx.doi.org/10.1090/S0002993902063670
PII:
S 00029939(02)063670
Keywords:
Lindel\"of,
nonreflection
Received by editor(s):
December 12, 2000
Received by editor(s) in revised form:
April 2, 2001
Published electronically:
March 13, 2002
Additional Notes:
Both authors were partially supported by the second author’s grant A7354 from the Natural Sciences and Engineering Research Council of Canada
Communicated by:
Alan Dow
Article copyright:
© Copyright 2002 American Mathematical Society
