A Lindelöf space with no Lindelöf subspace of size

Authors:
Piotr Koszmider and Franklin D. Tall

Journal:
Proc. Amer. Math. Soc. **130** (2002), 2777-2787

MSC (2000):
Primary 54A20, 54A25, 54A35; Secondary 03E35

DOI:
https://doi.org/10.1090/S0002-9939-02-06367-0

Published electronically:
March 13, 2002

MathSciNet review:
1900885

Full-text PDF

Abstract | References | Similar Articles | 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 set-theoretic 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 set-theoretic assumptions turn out to be necessary.

**[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. 603-607. 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 Set-Theory. Set-theoretic 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 set-theoretic topology*; North Holland, 1984. MR**85k:54001****[R]**M. E. Rudin;*Lectures on Set-Theoretic 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. 233-268. MR**82m:03062****[W]**W. Weiss;*Versions of Martin's Axiom*; in [KV] pp. 827 - 886. MR**86h:03088**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
54A20,
54A25,
54A35,
03E35

Retrieve articles in all journals with MSC (2000): 54A20, 54A25, 54A35, 03E35

Additional Information

**Piotr Koszmider**

Affiliation:
Departamento de Matemática, Universidade de São Paulo, Caixa Postal: 66281, São Paulo, SP, CEP: 05315-970, 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:
https://doi.org/10.1090/S0002-9939-02-06367-0

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 A-7354 from the Natural Sciences and Engineering Research Council of Canada

Communicated by:
Alan Dow

Article copyright:
© Copyright 2002
American Mathematical Society