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The least size of an infinite maximal almost disjoint family on omega. Zorn lemma extends an infinite disjoint family to a maximal one; thus $\mathfrak{a}\le 2^{{\aleph }_0}$. The bounding-to-almost-disjointness inequality supplies $\mathfrak{b}\le \mathfrak{a}$.

A subset of a partially ordered set on which the inherited order is total. The upper-bound hypothesis in Zorn lemma concerns all such subsets, not just countable increasing sequences.

An embedding of a subfield into an algebraically closed field extends over an algebraic extension: for a simple extension, map its generator to a root of the transported minimal polynomial, then iterate or use Zorn lemma. If the larger extension is normal over the fixed base, its image under the extended embedding is itself.

For a short exact sequence of sheaves with flasque sheaf kernel, every section of the quotient on any open set lifts to the middle sheaf on that set. Use the Zorn lemma to choose a maximal partial lift, with chains glued by the sheaf gluing axiom. A new local lift differs from the old one on their overlap by a kernel section. Extend that difference by flasqueness, adjust the new lift, and glue, contradicting maximality unless its domain is the entire open set. Consequently a quotient of two flasque sheaves is flasque. With an injective resolution of sheaves, the long exact sequence in sheaf cohomology and repeated dimension shifts prove that flasque sheaves have zero positive-degree cohomology.

A nonempty compact face of a convex set in a Hausdorff locally convex space contains an extreme point. Chains of compact faces have nonempty compact intersection. A minimal face supplied by the Zorn lemma must be a singleton, since a separating linear functional would otherwise expose a proper maximizer face. This is the existence step in the Krein-Milman theorem.

A minimal subsystem is a nonempty closed forward-invariant subset on which the restricted dynamical system is a minimal dynamical system. Every continuous map on a nonempty compact Hausdorff space has one: intersections of chains of nonempty closed invariant sets remain nonempty by compactness, so the Zorn lemma gives a minimal member.

For every commutative ring $R$,Every nilpotent element lies in every prime ideal. If $a$ is not nilpotent, an ideal maximal among those disjoint from $\{1,a,a^2,\dots \}$ is a prime ideal avoiding $a$, by the Zorn lemma and the maximality argument.