sets A and A×A. Trichotomy: If two sets are given, then either they have the same cardinality, or one has a smaller cardinality than the other. Given two non-empty sets, one has a surjection to the other. Every surjective function has a right inverse. The Cartesian product of any family of nonempty sets is nonempty. In other words, every family of nonempty sets has a choice function (i.e. a function which maps each of the nonempty sets to one of its elements). König's theorem: Colloquially, the sum of a sequence of cardinals is strictly less than the product of a sequence of larger cardinals. (The reason for the term "colloquially" is that the sum or product of a "sequence" of cardinals cannot itself be defined without some aspect of the axiom of choice.) Well-ordering theorem: Every set can be well-ordered. Consequently, every cardinal has an initial ordinal. Every element of a partially ordered set S is the minimal element of a well-ordered subset having no strict upper bound in S. Zorn's lemma: Every non-empty partially ordered set in which every chain (i.e., totally ordered subset) has an upper bound contains at least one maximal element. Hausdorff maximal principle: Every partially ordered set has a maximal chain. Equivalently, in any partially ordered set, every chain can be extended to a maximal chain. Tukey's lemma: Every non-empty collection of finite character has a maximal element with respect to inclusion. Antichain principle: Every partially ordered set has a maximal antichain. Equivalently, in any partially ordered set, every antichain can be extended to a maximal antichain. Abstract algebra Every vector space has a basis (i.e., a linearly independent spanning subset). In other words, vector spaces are equivalent to free modules. Krull's theorem: Every unital ring (other than the trivial ring) contains a maximal ideal. Equivalently, in any nontrivial unital ring, every ideal can be extended to a maximal ideal. For every non-empty set S there is a binary operation defined on S that gives it a group structure. (A cancellative binary operation is enough, see group structure and the axiom of choice.) Every free abelian group is projective. Baer's criterion: Every divisible abelian group is injective. Every set is a projective object in the category Set of sets. Functional analysis The closed unit ball of the dual of a normed vector space over the reals has an extreme point. Point-set topology The Cartesian product