Notre Dame Journal of Formal Logic

Thin Ultrafilters

O. Petrenko and I. V. Protasov


A free ultrafilter $\mathcal{U}$ on $\omega$ is called a $T$-point if, for every countable group $G$ of permutations of $\omega$, there exists $U\in\mathcal{U}$ such that, for each $g\in G$, the set $\{x\in U:gx\ne x, gx\in U\}$ is finite. We show that each $P$-point and each $Q$-point in $\omega^*$ is a $T$-point, and, under CH, construct a $T$-point, which is neither a $P$-point, nor a $Q$-point. A question whether $T$-points exist in ZFC is open.

Article information

Notre Dame J. Formal Logic, Volume 53, Number 1 (2012), 79-88.

First available in Project Euclid: 9 May 2012

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 54D35: Extensions of spaces (compactifications, supercompactifications, completions, etc.) 54D80: Special constructions of spaces (spaces of ultrafilters, etc.)

ultrafilter thin set $P$-point $Q$-point $T$-point


Petrenko, O.; Protasov, I. V. Thin Ultrafilters. Notre Dame J. Formal Logic 53 (2012), no. 1, 79--88. doi:10.1215/00294527-1626536.

Export citation


  • Chou, C., "On the size of the set of left invariant means on a semi-group", Proceedings of the American Mathematical Society, vol. 23 (1969), pp. 199–205.
  • Hart, K. P., and J. van Mill, "Open problems on $\beta\omega$", pp. 97–125 in Open Problems in Topology, North-Holland, Amsterdam, 1990.
  • Hindman, N., and D. Strauss, Algebra in the Stone-Čech Compactification. Theory and Applications, vol. 27 of de Gruyter Expositions in Mathematics, Walter de Gruyter & Co., Berlin, 1998.
  • Ketonen, J., "On the existence of $P$"-points in the Stone-Čech compactification of integers, Fundamenta Mathematicae, vol. 92 (1976), pp. 91–94.
  • Kunen, K., "Weak $P$"-points in ${\bf N}^{\ast} $, pp. 741–49 in Topology, Vol. II (Proceedings of the Fourth Colloquium, Budapest, 1978, edited by Á. Csa\' szár, vol. 23 of Colloquia Mathematica Societatis János Bolyai, North-Holland, Amsterdam, 1980.
  • Lutsenko, I., and I. V. Protasov, "Sparse, thin and other subsets of groups", International Journal of Algebra and Computation, vol. 19 (2009), pp. 491–510.
  • Lutsenko, I., "Thin systems of generators of groups", Algebra and Discrete Mathematics, vol. 9 (2010), pp. 108–14.
  • Mathias, A. R. D., "$0^{\#}$" and the $p$"-point problem, pp. 375–84 in Higher Set Theory (Proceedings of the Conference at Mathematisches Forschungsinstitut, Oberwolfach, 1977), edited by G. H. Müller and D. S. Scott, vol. 669 of Lecture Notes in Mathematics, Springer, Berlin, 1978.
  • van Mill, J., "An introduction to $\beta\omega$", pp. 503–67 in Handbook of Set-Theoretic Topology, North-Holland, Amsterdam, 1984.
  • Miller, A. W., "There are no $Q$"-points in Laver's model for the Borel conjecture, Proceedings of the American Mathematical Society, vol. 78 (1980), pp. 103–106.
  • Protasov, I. V., "Normal ball structures", Matematichn\=\i Stud\=\i ï. Prats\=\i Lriptsize $'$v\=\i vsriptsize $'$kogo Matematichnogo Tovaristva, vol. 20 (2003), pp. 3–16.
  • Protasov, I. V., "Coronas of balleans", Topology and its Applications, vol. 149 (2005), pp. 149–60.
  • Protasov, I. V., "Dynamical equivalences on $G^\ast$", Topology and its Applications, vol. 155 (2008), pp. 1394–1402.
  • Protasov, I. V., "Selective survey on subset combinatorics of groups", Ukraï nsriptsize $'$kiĭ Matematichniĭ V\=\i snik, vol. 7 (2010), pp. 220–57.
  • Protasov, I., and M. Zarichnyi, General Asymptology, vol. 12 of Mathematical Studies Monograph Series, VNTL Publishers, Lriptsize $'$viv, 2007.
  • Shelah, S., Proper Forcing, vol. 940 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1982.