Open covers and partition relations
HTML articles powered by AMS MathViewer
- 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).References
- James E. Baumgartner and Alan D. Taylor, Partition theorems and ultrafilters, Trans. Amer. Math. Soc. 241 (1978), 283–309. MR 491193, DOI 10.1090/S0002-9947-1978-0491193-1
- Fred Galvin, Indeterminacy of point-open games, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 26 (1978), no. 5, 445–449 (English, with Russian summary). MR 493925
- J. Gerlits and Zs. Nagy, Some properties of $C(X)$. I, Topology Appl. 14 (1982), no. 2, 151–161. MR 667661, DOI 10.1016/0166-8641(82)90065-7
- W. Hurewicz, Über die Verallgemeinerung des Borelschen Theorems, Mathematische Zeitschrift 24 (1925), 401 – 421.
- W. Just, A.W. Miller, M. Scheepers and P.J. Szeptycki, Combinatorics of open covers (II), Topology and its Applications 73 (1996), 241 – 266.
- K. Menger, Einige Überdeckungssätze der Punktmengenlehre, Sitzungsberichte der Wiener Akademie Abt. 2a, Mathematik, Astronomie, Physik, Meteorologie und Mechanik 133 (1924), 421 – 444.
- Janusz Pawlikowski, Undetermined sets of point-open games, Fund. Math. 144 (1994), no. 3, 279–285. MR 1279482
- F.P. Ramsey, On a problem of formal logic, Proceedings of the London Mathematical Society 30 (1930), 264 – 286.
- F. Rothberger, Eine Verschärfung der Eigenschaft $\mathsf {C}$, Fundamenta Mathematicae 30 (1938), 50 – 55.
- Marion Scheepers, Combinatorics of open covers. I. Ramsey theory, Topology Appl. 69 (1996), no. 1, 31–62. MR 1378387, DOI 10.1016/0166-8641(95)00067-4
- M. Scheepers, Rothberger’s property and partition relations, The Journal of Symbolic Logic 62 (1997), 976–980.
- Marion Scheepers, Open covers and the square bracket partition relation, Proc. Amer. Math. Soc. 125 (1997), no. 9, 2719–2724. MR 1396995, DOI 10.1090/S0002-9939-97-03898-7
- Rastislav Telgársky, Remarks on a game of Choquet, Colloq. Math. 51 (1987), 365–372. MR 891306, DOI 10.4064/cm-51-1-365-372
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