Hausdorff's Maximal Principle/Formulation 2

Theorem

Let $A$ be a non-empty set of sets.

Let $S$ be the set of all chain of sets of $A$ (ordered under the subset relation).

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

Also known as

Hausdorff's Maximal Principle is also known as the Hausdorff Maximal Principle.

Some sources call it the Hausdorff Maximality Principle or the Hausdorff Maximality Theorem.

Also see

• Results about Hausdorff's maximal principle can be found here.

Source of Name

This entry was named for Felix Hausdorff.

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: Proposition $5.8$