Open covers and the square bracket

partition relation

Author:
Marion Scheepers

Journal:
Proc. Amer. Math. Soc. **125** (1997), 2719-2724

MSC (1991):
Primary 03E05

DOI:
https://doi.org/10.1090/S0002-9939-97-03898-7

MathSciNet review:
1396995

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: An open cover of an infinite separable metric space is an -cover of if and for every finite subset of there is a such that . Let be the collection of -covers of . We show that the partition relation holds if, and only if, the partition relation holds.

**[1]**J.E. Baumgartner and A.D. Taylor,*Partition theorems and ultrafilters*,**Transactions of the American Mathematical Society**241 (1978), 283 - 309. MR**58:10458****[2]**A.R. Blass,*Ultrafilter mappings and their Dedekind cuts*,**Transactions of the American Mathematical Society**188 (1974), 327 - 340. MR**50:4310****[3]**D. Booth,*Ultrafilters on a countable set*,**Annals of Mathematical Logic**2 (1970/71), 1 - 24. MR**43:3104****[4]**D. Devlin,*Some partition theorems and ultrafilters on*, Ph. D. thesis, Dartmouth College (1979)**[5]**P. Erdös, A. Hajnal and R. Rado,*Partition relations for cardinal numbers*,**Acta Math. Acad. Sci. Hungar.**16 (1965), 93-196. MR**34:2475****[6]**W. Just, A.W. Miller, M. Scheepers and P.J. Szeptycki,*Combinatorics of Open Covers II*,**Topology and its Applications**73 (1996), 241-266.**[7]**F.P. Ramsey,*On a problem of formal logic*,**Proceedings of the London Mathematical Society**30 (1930), 264 - 286.**[8]**F. Rothberger,*Eine Verschärfung der Eigenschaft*,*C***Fundamenta Mathematicae**30 (1938), 50 - 55.**[9]**M. Sheepers,*Combinatorics of Open Covers I: Ramsey theory*,**Topology and its Applications**69 (1996), 31-62. CMP**96:09****[10]**M. Scheepers,*Rothberger's property and partition relations*,**The Journal of Symbolic Logic**, to appear.**[11]**N.H. Williams,*Combinatorial Set Theory*,**North-Holland**(1980).

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (1991):
03E05

Retrieve articles in all journals with MSC (1991): 03E05

Additional Information

**Marion Scheepers**

Affiliation:
Department of Mathematics and Computer Science Boise State University Boise, Idaho 83725

Email:
marion@math.idbsu.edu

DOI:
https://doi.org/10.1090/S0002-9939-97-03898-7

Keywords:
Rothberger property,
square bracket partition relation,
Ramsey's theorem

Received by editor(s):
October 13, 1995

Received by editor(s) in revised form:
April 11, 1996

Additional Notes:
The author was supported by NSF grant DMS 95 - 05375

Communicated by:
Andreas R. Blass

Article copyright:
© Copyright 1997
American Mathematical Society