Hausdorff gap

Template:Inline 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 Template:Harvs. 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 $\lim (g(n)-f(n))=+\mathrm{\infty }$.

If $X$ is a poset and $\kappa$ and $\lambda$ are cardinals, then a $(\kappa ,\lambda )$-pregap in $X$ is a set of elements $f_{\alpha }$ for $\alpha \in \kappa$ and a set of elements $g_{\beta }$ for $\beta \in \lambda$ 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 Template:Harvtxt 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 $\kappa \ge {\omega }_2$.

References

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External links

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