The axiom of choice is an axiom usually added to the Zermelo-Fraenkel theory of sets, it states that given a set $I$ and $a_i$, for $i\in I$, non-empty sets there exists a function $f\colon I\to \bigcup a_i$ such that $f(i)\in a_i$ such a function is called a choice function since it chooses one element $f(i)$ from each of the sets $a_i$.
The axiom of choice is equivalent to the statement that every set can be well-ordered, as well to the Zorn's lemma which asserts that if a partial order has the property that every chain is bounded from above, then there is a maximal element.
The axiom of choice is generally independent of the ZF theory, and if ZF is consistent then it is consistent with both the axiom of choice as well its negation.
Some theorems which follow from the axiom of choice:
- The product of compact spaces is compact (equivalent)
- Every surjective map has a right inverse (equivalent)
- Every vector space has a basis (equivalent)
- Countable union of countable sets is countable
- Every infinite set has a countable subset
- Every field has an algebraic closure
- There are sets of real numbers which are not Lebesgue measurable
Most of the mathematicians nowadays assume the axiom of choice when they deal with their mathematics. Mostly because without it infinite processes in mathematics become much harder to handle, and one can construct models without the axiom of choice in which continuity of sequences is not equivalent to $\varepsilon-\delta$ continuity, or some fields have no algebraic closure.