Hausdorff gap explained

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.

Definition

Let

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

If

X is a poset and \kappa and λ are cardinals, then a (\kappa, λ ) -pregap in X is a set of elements f _{\alpha} for \alpha\in\kappa and a set of elements g _{\beta} for \beta\in λ such that:

• The transfinite sequence

f is strictly increasing;

• The transfinite sequence

g is strictly decreasing;

• Every element of the sequence

f is less than every element of the sequence g .

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

• There is no element

h greater than all elements of f and less than all elements of g .

A Hausdorff gap is a

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

There are some variations of these definitions, with the ordered set

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

Existence

It is possible to prove in ZFC that there exist Hausdorff gaps and

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