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 . The existence of Hausdorff gaps shows that the partially ordered set of possible growth rates of sequences is not complete.
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:
A pregap is called a gap if it satisfies the additional condition:
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 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 .
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 .