# Axiom:Axiom of Choice/Formulation 2

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## Axiom

Let $\left \langle {X_i} \right \rangle_{i \mathop \in I}$ be an indexed family of sets all of which are non-empty, indexed by $I$ which is also non-empty.

Then there exists an indexed family $\left \langle {x_i} \right \rangle_{i \mathop \in I}$ such that:

- $\forall i \in I: x_i \in X_i$

That is, the Cartesian product of a non-empty family of sets which are non-empty is itself non-empty.

## Also see

- Equivalence of Versions of Axiom of Choice
- Results about
**the Axiom of Choice**can be found here.

## Sources

- 1960: Paul R. Halmos:
*Naive Set Theory*... (previous) ... (next): $\S 15$: The Axiom of Choice - 1964: Steven A. Gaal:
*Point Set Topology*... (previous) ... (next): Introduction to Set Theory: $3$. The Axiom of Choice and Its Equivalents