Axiom:Axiom of Dependent Choice

Axiom
Let $\mathcal R$ be a binary relation on a non-empty set $S$.

Suppose that:
 * $\forall a \in S: \exists b \in S: a \mathop {\mathcal R} b$

that is, that $\mathcal R$ is a left-total relation.

The axiom of dependent choice states that there exists a sequence $\left\langle{x_n}\right\rangle_{n \in \N}$ in $S$ such that:
 * $\forall n \in \N: x_n \mathop {\mathcal R} x_{n+1}$

This axiom can be abbreviated ADC or simply DC.

Also see
This axiom is a weaker form of the axiom of choice, as shown in Axiom of Choice Implies Axiom of Dependent Choice.

This axiom is also a stronger form of the axiom of countable choice, as shown in Axiom of Dependent Choice Implies Axiom of Countable Choice.