Real Analysis Exchange

A Topological Interpretation of t

Boaz Tsaban

Full-text: Open access

Abstract

Hurewicz found connections between some topological notions and the combinatorial cardinals $\mathfrak{b}$ and $\mathfrak{d}$. Recɫaw gave topological meaning to the definition of the cardinal $\mathfrak{p}$. We extend the picture with a topological interpretation of the cardinal $\mathfrak{t}$. We compare our notion to the one related to $\mathfrak{p}$, and to some other classical notions. This sheds new light on the famous open problem whether $\mathfrak{p}=\mathfrak{t}$.

Article information

Source
Real Anal. Exchange, Volume 24, Number 1 (1998), 391-404.

Dates
First available in Project Euclid: 23 March 2011

Permanent link to this document
https://projecteuclid.org/euclid.rae/1300906035

Mathematical Reviews number (MathSciNet)
MR1691758

Zentralblatt MATH identifier
0938.03071

Subjects
Primary: 03E50: Continuum hypothesis and Martin's axiom [See also 03E57] 03E10: Ordinal and cardinal numbers 04A15

Keywords
$\p$ $\ft$ $\g$-cover small sets $\lambda$-sets infinitary combinatorics

Citation

Tsaban, Boaz. A Topological Interpretation of t. Real Anal. Exchange 24 (1998), no. 1, 391--404. https://projecteuclid.org/euclid.rae/1300906035


Export citation

References

  • T. Bartoszyński and H. Judah, Borel images of sets of reals, Real Analysis Exchange 20 (1994/5), 536–558.
  • J. Brendle, Generic constructions of small sets of reals, Topology Appl. 71 (1996), 125–147.
  • P. Daniels, Pixeley-Roy spaces over subsets of the reals, Topology Appl. 29 (1988), 93–106.
  • E.K. van Douwen, The integers and topology, in: Handbook of Set Theoretic Topology (K. Kunen and J. Vaughan, Eds.), North-Holland, Amsterdam, 1984, 111–167.
  • F. Galvin and A.W. Miller, $\g$-sets and other singular sets of real numbers, Topology Appl. 17 (1984), 145–155.
  • J. Gerlits and Zs. Nagy, Some properties of $C(X)$, I, Topology Appl. 14 (1982), 151–161.
  • W. Hurewicz, Über eine Verallgemeinerung des Borelschen Theorems, Math. Z. 24 (1925), 401–421.
  • ––––, Über Folgen stetiger Funktionen, Fund. Math. 9 (1927), 193–204.
  • W. Just, A.W. Miller, M. Scheepers, and P. J. Szeptycki, The combinatorics of open covers II, Topology Appl. 73 (1996), 241–266.
  • A.W. Miller, Special subsets of the real line, in: Handbook of Set Theoretic Topology (K. Kunen and J. Vaughan, Eds.), North-Holland, Amsterdam, 1984, 201–233.
  • Y.N. Moschovakis, Descriptive Set Theory, North-Holland, Amsterdam, 1980.
  • J. Pawlikowski and I. Recław, Parametrized Cichon's diagram and small sets, Fund. Math. 147 (1995), 135–155.
  • I. Recław, Every Luzin set is undetermined in point-open game, Fund. Math. 144 (1994), 43–54.
  • F. Rothberger, Sur un ensemble toujours de premiére categorie qui est depourvu de la propriété $\lambda$, Fund. Math. 32 (1939), 294–300.
  • ––––, On some problems of Hausdorff and of Sierpińskii, Fund. Math. 35 (1948), 29–46.
  • ––––, Sur les familles indénombrables de suites de nombres naturels et les problèmes concernant la propriété C, Proc. Cambr. Phil. Soc. 37 (1941), 109–126.
  • Z. Shuguo, Relations among cardinal invariants, in: Proceedings of Chinese Conference on Pure and Applied Logic (ed. Z. Jinwen), Beijing 1992, 145–147.
  • S. Todorčević, Aronszajn Orderings, Publ. Inst. Math. 57 (1995), 29–46.