# Restricted Tukey-Teichmüller Theorem

## Theorem

### Weak Form

Let $X$ be a set.

Let $\mathcal A$ be a non-empty set of subsets of $X$.

Let $'$ be a unary operation on $X$.

Let $\mathcal A$ have finite character.

For all $A \in \mathcal A$ and all $x \in X$, let either:

$A \cup \set x \in \mathcal A$

or:

$A \cup \set {x'} \in \mathcal A$

Then there exists a $B \in \mathcal A$ such that for all $x \in X$, either $x \in B$ or $x' \in B$.

### Strong Form

Let $X$ be a set.

Let $\mathcal A$ be a non-empty set of subsets of $X$.

Let $'$ be a unary operation on $X$.

Let $\mathcal A$ have finite character.

For all $A \in \mathcal A$ and all $x \in X$, let either:

$A \cup \left\{ {x}\right\} \in \mathcal A$

or:

$A \cup \left\{ {x'}\right\} \in \mathcal A$

Then for each $A \in \mathcal A$ there exists a $C \in \mathcal A$ such that:

$A \subseteq C$

and:

for all $x \in X$, either $x \in C$ or $x' \in C$.

## Source of Name

This entry was named for John Wilder Tukey and Oswald Teichmüller.