Let ${\displaystyle \omega ^{\omega }}$  be the set of all sequences of non-negative integers, and define ${\displaystyle f<g}$  to mean ${\displaystyle \lim \left(g(n)-f(n)\right)=+\infty }$ .

If ${\displaystyle X}$  is a poset and ${\displaystyle \kappa }$  and ${\displaystyle \lambda }$  are cardinals, then a ${\displaystyle (\kappa ,\lambda )}$ -pregap in ${\displaystyle X}$  is a set of elements ${\displaystyle f_{\alpha }}$  for ${\displaystyle \alpha \in \kappa }$  and a set of elements ${\displaystyle g_{\beta }}$  for ${\displaystyle \beta \in \lambda }$  such that:

• The transfinite sequence ${\displaystyle f}$  is strictly increasing;
• The transfinite sequence ${\displaystyle g}$  is strictly decreasing;
• Every element of the sequence ${\displaystyle f}$  is less than every element of the sequence ${\displaystyle g}$ .

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

• There is no element ${\displaystyle h}$  greater than all elements of ${\displaystyle f}$  and less than all elements of ${\displaystyle g}$ .

A Hausdorff gap is a ${\displaystyle (\omega _{1},\omega _{1})}$ -gap in ${\displaystyle \omega ^{\omega }}$  such that for every countable ordinal ${\displaystyle \alpha }$  and every natural number ${\displaystyle n}$  there are only a finite number of ${\displaystyle \beta }$  less than ${\displaystyle \alpha }$  such that for all ${\displaystyle k>n}$  we have ${\displaystyle f_{\alpha }(k)<g_{\beta }(k)}$ .

There are some variations of these definitions, with the ordered set ${\displaystyle \omega ^{\omega }}$  replaced by a similar set. For example, one can redefine ${\displaystyle f<g}$  to mean ${\displaystyle f(n)<g(n)}$  for all but finitely many ${\displaystyle n}$ . Another variation introduced by Hausdorff (1936) is to replace ${\displaystyle \omega ^{\omega }}$  by the set of all subsets of ${\displaystyle \omega }$ , with the order given by ${\displaystyle A<B}$  if ${\displaystyle A}$  has only finitely many elements not in ${\displaystyle B}$  but ${\displaystyle B}$  has infinitely many elements not in ${\displaystyle A}$ .

It is possible to prove in ZFC that there exist Hausdorff gaps and ${\displaystyle (b,\omega )}$ -gaps where ${\displaystyle b}$  is the cardinality of the smallest unbounded set in ${\displaystyle \omega ^{\omega }}$ , and that there are no ${\displaystyle (\omega ,\omega )}$ -gaps. The stronger open coloring axiom can rule out all types of gaps except Hausdorff gaps and those of type ${\displaystyle (\kappa ,\omega )}$  with ${\displaystyle \kappa \geq \omega _{2}}$ .

• 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
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• Scheepers, Marion (1993), "Gaps in ω^ω", 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

• "Hausdorff gap", Encyclopedia of Mathematics, EMS Press, 2001 [1994]