In mathematics, a Hausdorff gap consists roughly of two collections of sequences of integers, such that there is no sequence lying between the two collections. The first example was found by Hausdorff (1909). The existence of Hausdorff gaps shows that the partially ordered set of possible growth rates of sequences is not complete.

Definition

edit

Let   be the set of all sequences of non-negative integers, and define   to mean  .

If   is a poset and   and   are cardinals, then a  -pregap in   is a set of elements   for   and a set of elements   for   such that:

  • The transfinite sequence   is strictly increasing;
  • The transfinite sequence   is strictly decreasing;
  • Every element of the sequence   is less than every element of the sequence  .

A pregap is called a gap if it satisfies the additional condition:

  • There is no element   greater than all elements of   and less than all elements of  .

A Hausdorff gap is a  -gap in   such that for every countable ordinal   and every natural number   there are only a finite number of   less than   such that for all   we have  .

There are some variations of these definitions, with the ordered set   replaced by a similar set. For example, one can redefine   to mean   for all but finitely many  . Another variation introduced by Hausdorff (1936) is to replace   by the set of all subsets of  , with the order given by   if   has only finitely many elements not in   but   has infinitely many elements not in  .

Existence

edit

It is possible to prove in ZFC that there exist Hausdorff gaps and  -gaps where   is the cardinality of the smallest unbounded set in  , and that there are no  -gaps. The stronger open coloring axiom can rule out all types of gaps except Hausdorff gaps and those of type   with  .

References

edit
  • Carotenuto, Gemma (2013), An introduction to OCA (PDF), notes on lectures by Matteo Viale
  • Ryszard, Frankiewicz; Paweł, Zbierski (1994), Hausdorff gaps and limits, Studies in Logic and the Foundations of Mathematics, vol. 132, Amsterdam: North-Holland Publishing Co., ISBN 0-444-89490-X, MR 1311476
  • Hausdorff, F. (1909), Die Graduierung nach dem Endverlauf, Abhandlungen der Königlich Sächsischen Gesellschaft der Wissenschaften zu Leipzig, vol. 31, B. G. Teubner, pp. 296–334
  • Hausdorff, F. (1936), "Summen von ℵ1 Mengen" (PDF), Fundamenta Mathematicae, 26 (1), Institute of Mathematics Polish Academy of Sciences: 241–255, doi:10.4064/fm-26-1-241-255, ISSN 0016-2736
  • Scheepers, Marion (1993), "Gaps in ωω", in Judah, Haim (ed.), Set theory of the reals (Ramat Gan, 1991), Israel Math. Conf. Proc., vol. 6, Ramat Gan: Bar-Ilan Univ., pp. 439–561, ISBN 978-9996302800, MR 1234288
edit