Tukey's Lemma/Formulation 2

Theorem

Let $S$ be a non-empty set of finite character.

Then every element of $S$ is a subset of a maximal element of $S$ under the subset relation.

Proof

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Also known as

Tukey's Lemma is still occasionally found with the name of Teichmüller attached to it, but this is dying out.

Source of Name

This entry was named for John Wilder Tukey.

Sources

• 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $4$: Superinduction, Well Ordering and Choice: Part $\text {II}$ -- Maximal principles: $\S 5$ Maximal principles