Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Open covers and partition relations

Author: Marion Scheepers
Journal: Proc. Amer. Math. Soc. 127 (1999), 577-581
MSC (1991): Primary 03E05, 05D10
MathSciNet review: 1458262
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: An open cover of a topological space is said to be an $\omega$-cover if there is for each finite subset of the space a member of the cover which contains the finite set, but the space itself is not a member of the cover. We prove theorems which imply that a set $X$ of real numbers has Rothberger's property $\textsf{C}^{\prime\prime}$ if, and only if, for each positive integer $k$, for each $\omega$-cover $\mathcal{U}$ of $X$, and for each function $f:[\mathcal{U}]^2\rightarrow\{1,\dots,k\}$ from the two-element subsets of $\mathcal{U}$, there is a subset $\mathcal{V}$ of $\mathcal{U}$ such that $f$ is constant on $[\mathcal{V}]^2$, and each element of $X$ belongs to infinitely many elements of $\mathcal{V}$ (Theorem 1). A similar characterization is given of Menger's property for sets of real numbers (Theorem 6).

References [Enhancements On Off] (What's this?)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 03E05, 05D10

Retrieve articles in all journals with MSC (1991): 03E05, 05D10

Additional Information

Marion Scheepers
Affiliation: Department of Mathematics, Boise State University, Boise, Idaho 83725

Keywords: Ramsey's theorem, Rothberger's property, Menger's property, infinite game, partition relation
Received by editor(s): April 15, 1996
Received by editor(s) in revised form: May 16, 1997
Additional Notes: The author’s research was funded in part by NSF grant DMS 95-05375
Communicated by: Andreas R. Blass
Article copyright: © Copyright 1999 American Mathematical Society