Zornlemma

Zorn's lemma is a result in set theory and order theory that provides a sufficient condition for the existence of maximal elements in partially ordered sets. It states that if every chain (totally ordered subset) in a nonempty partially ordered set has an upper bound within the set, then the set contains at least one maximal element. This lemma is frequently used in algebra, topology, and functional analysis to guarantee the existence of structures such as bases for vector spaces, maximal ideals, and extensions of functions.

The lemma is named after M. C. Zorn, who first published it in 1935. It is equivalent, under the usual axioms of Zermelo–Fraenkel set theory, to several other statements often used as alternatives to the Axiom of Choice: the Well‑Ordering Theorem, Zorn's lemma itself, and the Hausdorff Maximal Principle. Because of these equivalences, many mathematicians accept Zorn's lemma as a convenient form of the Axiom of Choice, while others treat it as a theorem derived from that axiom.

Proofs of Zorn's lemma usually proceed by constructing a chain that absolutely cannot be extended, often using transfinite induction or a maximality argument. The lemma is applied, for example, to prove that every vector space has a basis, that every nonzero module over a ring contains a simple submodule, or that a topological space that satisfies the compactness condition has a minimal closed set containing a given point.

Mathematical usage of Zorn's lemma is widespread in proofs where explicit construction of an element is difficult or impossible, allowing the existence of such elements to be guaranteed purely by order‑theoretic properties.