Axiom:Axiom of Countable Choice

Axiom

Form 1

Let $\left\langle{S_n}\right\rangle_{n \mathop \in \N}$ be a sequence of non-empty sets.

The axiom of countable choice states that there exists a sequence:

$\left\langle{x_n}\right\rangle_{n \mathop \in \N}$

such that $x_n \in S_n$ for all $n \in \N$.

Form 2

Let $S$ be a countable set of non-empty sets.

Then $S$ has a choice function.

Also known as

This axiom can be abbreviated $\mathrm{ACC}$, $\mathrm{CC}$, $\mathrm{AC}_\omega$, or $\mathrm{AC}_\N$.