Definition:Set Product/Family of Sets

Definition
Let $\left\langle{S_i}\right\rangle_{i \mathop \in I}$ be an indexed family of sets.

Let $P$ be a set and let $\left\langle{\phi_i}\right\rangle_{i \mathop \in I}$ be an indexed family of mappings $\phi_i: P \to S_i$ for all $i \in I$ such that:


 * For all sets $X$ and all indexed families $\left\langle{f_i}\right\rangle_{i \mathop \in I}$ of mappings $f_i: X \to S_i$ there exists a unique mapping $h: X \to P$ such that:
 * $\forall i \in I: \phi_i \circ h = f_i$


 * that is, such that for all $i \in I$:


 * $\begin{xy}\xymatrix@+1em@L+3px{

X \ar@{-->}[d]_*+{h} \ar[dr]^*+{f_i}

\\ P \ar[r]_*{\phi_i} & S_i }\end{xy}$


 * is a commutative diagram.

Then $P$, together with the family of mappings $\left\langle{\phi_i}\right\rangle_{i \mathop \in I}$, is called a product of (the family) $\left\langle{S_i}\right\rangle_{i \mathop \in I}$.

This product of $\left\langle{S_i}\right\rangle_{i \mathop \in I}$ can be denoted $\left({P, \left\langle{\phi_i}\right\rangle_{i \mathop \in I}}\right)$.

Also see

 * Definition:Cartesian Product of Family
 * Definition:Product (Category Theory)/General Definition