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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
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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).

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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

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