Axiom:Axiom of Choice/Formulation 2

Axiom

Let $\family {X_i}_{i \mathop \in I}$ 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 $\family {x_i}_{i \mathop \in I}$ 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 Formulations 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
• 1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): Chapter $1$: Theory of Sets: $\S 10$: Arbitrary Products