Restricted Tukey's Theorem

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Theorem

Weak Form

Let $X$ be a set.

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

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

Let $\AA$ have finite character.

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

$A \cup \set x \in \AA$

or:

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


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


Strong Form

Let $X$ be a set.

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

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


Let $\AA$ have finite character.

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

$A \cup \set x \in \AA$

or:

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


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

$A \subseteq C$

and:

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


Variation 1

Restricted Tukey's Theorem/Variation 1

Variation 2

Restricted Tukey's Theorem/Variation 2

Source of Name

This entry was named for John Wilder Tukey.


Sources