A sequential property of $\mathsf {C}_p(X)$ and a covering property of Hurewicz
HTML articles powered by AMS MathViewer
- by Marion Scheepers
- Proc. Amer. Math. Soc. 125 (1997), 2789-2795
- DOI: https://doi.org/10.1090/S0002-9939-97-03897-5
- PDF | Request permission
Abstract:
$\mathsf {C}(X)$ has the monotonic sequence selection property if there is for each $f$, and for every sequence $(\sigma _n:n<\omega )$ where for each $n$ $\sigma _n$ is a sequence converging pointwise monotonically to $f$, a sequence $(f_n:n<\omega )$ such that for each $n$ $f_n$ is a term of $\sigma _n$, and $(f_n:n<\omega )$ converges pointwise to $f$. We prove a theorem which implies for metric spaces $X$ that $\mathsf {C}(X)$ has the monotonic sequence selection property if, and only if, $X$ has a covering property of Hurewicz.References
- A. V. Arkhangel′skiĭ, Hurewicz spaces, analytic sets and fan tightness of function spaces, Dokl. Akad. Nauk SSSR 287 (1986), no. 3, 525–528 (Russian). MR 837289
- Andreas Blass and Thomas Jech, On the Egoroff property of pointwise convergent sequences of functions, Proc. Amer. Math. Soc. 98 (1986), no. 3, 524–526. MR 857955, DOI 10.1090/S0002-9939-1986-0857955-3
- 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. Hurewicz, Über Folgen stetiger Funktionen, Fundamenta Mathematicae 9 (1927), 193 – 204.
- W. Just, A.W. Miller, M. Scheepers, P.J. Szeptycki, Combinatorics of open covers (II), Topology and its Applications 73 (1996), 241–266.
- K. Menger, Einige Überdeckungssätze der Punktmengenlehre, Sitzungsberichte. Abt. 2a, Mathematik, Astronomie, Physik, Meteorologie und Mechanik (Wiener Akademie) 133 (1924), 421 - 444.
- Arnold W. Miller, Special subsets of the real line, Handbook of set-theoretic topology, North-Holland, Amsterdam, 1984, pp. 201–233. MR 776624
- F. Rothberger, Eine Verschärfung der Eigenschaft $\mathsf { C}''$, Fundamenta Mathematicae 30 (1938), 50 – 55.
- Leonard Eugene Dickson, New First Course in the Theory of Equations, John Wiley & Sons, Inc., New York, 1939. MR 0000002
- Masami Sakai, Property C′′and function spaces, Proc. Amer. Math. Soc. 104 (1988), no. 3, 917–919. MR 964873, DOI 10.1090/S0002-9939-1988-0964873-0
- M. Scheepers, Combinatorics of open covers I: Ramsey theory, Topology and its Applications, 69 (1996), no. 1, 31–62.
- Eric K. van Douwen, The integers and topology, Handbook of set-theoretic topology, North-Holland, Amsterdam, 1984, pp. 111–167. MR 776622
Bibliographic 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): December 15, 1995
- Received by editor(s) in revised form: April 11, 1996
- Additional Notes: The author’s research was supported in part by NSF grant DMS 95 - 05375.
- Communicated by: Andreas R. Blass
- © Copyright 1997 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 125 (1997), 2789-2795
- MSC (1991): Primary 54E99
- DOI: https://doi.org/10.1090/S0002-9939-97-03897-5
- MathSciNet review: 1396994