Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

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



Gaps in $({\mathcal P}(\omega ),\subset ^*)$ and $(\omega ^{\omega },\le ^*)$

Author: Zoran Spasojevic
Journal: Proc. Amer. Math. Soc. 124 (1996), 3857-3865
MSC (1991): Primary 03E05
MathSciNet review: 1327045
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: For a partial order $(P,\le _P)$, let $\Gamma (P,\le _P)$ denote the statement that for every $\le _P$-increasing $\omega _1$-sequence $a\subseteq P$ there is a $\le _P$-decreasing $\omega _1$-sequence $b\subseteq P$ on top of $a$ such that $(a,b)$ is an $(\omega _1,\omega _1)$-gap in $P$. The main result of this paper is that $\mathfrak t>\omega _1\leftrightarrow \Gamma(\mathcal P(\omega ),\subset ^*)\leftrightarrow \Gamma (\omega ^\omega ,\le ^*)$. It is also shown, as a corollary, that $\Gamma (\omega ^\omega ,\le ^*)\to \mathfrak b>\omega _1$ but $\mathfrak b>\omega _1\not \to\Gamma (\omega ^\omega ,\le ^*)$.

References [Enhancements On Off] (What's this?)

  • 1. Murray G. Bell, On the combinatorial principle 𝑃(𝔠), Fund. Math. 114 (1981), no. 2, 149–157. MR 643555
  • 2. H. G. Dales and W. H. Woodin, An introduction to independence for analysts, London Mathematical Society Lecture Note Series, vol. 115, Cambridge University Press, Cambridge, 1987. MR 942216
  • 3. F. Hausdorff, Die Graduierung nach dem Endverlauf, Abhandlungen der Königlich Sächsischen Gesellschaft der Wissenschaften, Mathematisch-Physische Klasse 31 (1909), 296-334.
  • 4. F. Hausdorff, Summen von $\aleph _1$ Mengen, Fund. Math. 26 (1936), 241-255.
  • 5. K. Kunen, Set Theory, An introduction to independence proofs, North-Holland, Amsterdam, 1980.
  • 6. F. Rothberger, Sur les familles indécombrables de suites de nombres naturels et les problèmes concernant la propriété C, Proc. Cambridge Philos. Soc. 37 (1941), 109-126. MR 2:352a
  • 7. Marion Scheepers, Gaps in 𝜔^{𝜔}, Set theory of the reals (Ramat Gan, 1991) Israel Math. Conf. Proc., vol. 6, Bar-Ilan Univ., Ramat Gan, 1993, pp. 439–561. MR 1234288
  • 8. R. C. Solomon, Families of sets and functions, Czechoslovak Math. J. 27(102) (1977), no. 4, 556–559. MR 0457218
  • 9. Zoran Spasojević, Some results on gaps, Topology Appl. 56 (1994), no. 2, 129–139. MR 1266138,
  • 10. Stevo Todorčević, Partition problems in topology, Contemporary Mathematics, vol. 84, American Mathematical Society, Providence, RI, 1989. MR 980949
  • 11. Eric K. van Douwen, The integers and topology, Handbook of set-theoretic topology, North-Holland, Amsterdam, 1984, pp. 111–167. MR 776622

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 03E05

Retrieve articles in all journals with MSC (1991): 03E05

Additional Information

Zoran Spasojevic
Affiliation: Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706
Address at time of publication: Institute of Mathematics, Hebrew University of Jerusalem, 91904 Jerusalem, Israel

Received by editor(s): September 6, 1994
Received by editor(s) in revised form: March 27, 1995
Communicated by: Andreas R. Blass
Article copyright: © Copyright 1996 American Mathematical Society