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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Open covers and partition relations
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by Marion Scheepers PDF
Proc. Amer. Math. Soc. 127 (1999), 577-581 Request permission

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
  • MR Author ID: 293243
  • Email: marion@math.idbsu.edu
  • 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
  • © Copyright 1999 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 127 (1999), 577-581
  • MSC (1991): Primary 03E05, 05D10
  • DOI: https://doi.org/10.1090/S0002-9939-99-04513-X
  • MathSciNet review: 1458262