Hausdorff gap

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 ${\displaystyle A}$.