Kuratowski's free set theorem

From Wikipedia, the free encyclopedia

Kuratowski's free set theorem, named after Kazimierz Kuratowski, is a result of set theory, an area of mathematics. It is a result which has been largely forgotten for almost 50 years, but has been applied recently in solving several lattice theory problems.

Denote by $[X] < ω$ the set of all finite subsets of a set $X$ . Likewise, for a positive integer $n$ , denote by $[X] n$ the set of all $n$ -elements subsets of $X$ . For a mapping $\Phi\colon[X]^n\to[X]^{<\omega}$ , we say that a subset $U$ of $X$ is free (with respect to $Φ$ ), if $u\notin\Phi(V)$ , for any $n$ -element subset $V$ of $U$ and any $u\in U\setminus V$ . Kuratowski published in 1951 the following result, which characterizes the infinite cardinals of the form $\aleph_n$ .

The theorem states the following. Let $n$ be a positive integer and let $X$ be a set. Then the cardinality of $X$ is greater than or equal to $\aleph_n$ if and only if for every mapping $Φ$ from $[X] n$ to $[X] < ω$ , there exists an $(n + 1)$ -element free subset of $X$ with respect to $Φ$ .