Axiom of Choice

Axiom of Choice

Suppose the sets $A_i$ are all nonempty for $i\in I$. Then there exists a function $f$ such that $\text{for all}i\in I,f(i)\in A_i.$ Equivalently, the product set ${\prod }_{i\in I}{A}_i:=\left\{f\mid f:I\to {\bigcup }_{i\in I}{A}_i,f(i)\in A_i\text{ if }i\in I\right\}$ is nonempty.

Remark

If $I$ is finite, say $I=\{1,...,n\}$, then up to notational changes this is just stating$A_1\times \cdots \times A_n\ne \varnothing$ This does not require the Axiom of Choice. It essentially follows from the much more basic and concrete Pairing Axiom applied the appropriate number of times,