Axiom of choice

In mathematics, the Axiom of Choice or AC is a fundamental principle in set theory which states that it is possible to choose an element out of each of infinitely many sets simultaneously. The validity of the axiom is not universally accepted among mathematicians and Kurt Gödel showed that it was independent of the other axioms of set theory.

The axiom states that if ${\mathcal {A}}$ is a family of non-empty sets, there is a choice function $f:{\mathcal {A}}\rightarrow \cup {\mathcal {A}}$ such that for each $A\in {\mathcal {A}}$ we have $f(A)\in A$: that is, $f$ "chooses" an element of each member of the family ${\mathcal {A}}$.

A closely related formulation of the axiom is that the Cartesian product of any family of non-empty sets is again non-empty.

Equivalent formulations

There are a number of statements equivalent to the Axiom of Choice.

• Zorn's Lemma: If every chain in a partially ordered set has an upper bound, then the set has a maximal element.
• The Well-ordering Principle: Every set can be well-ordered.
• Tukey's Lemma: Every non-empty system of finite character has a maximal element.
• Zermelo's Postulate: If ${\mathcal {A}}$ is a family of non-empty sets, there is a set $C$ such that $C\cap A$ has exactly one element for each $A\in {\mathcal {A}}$.
• Tychonov's Theorem: The product of a family of non-empty compact topological spaces is compact in the product topology.