Definition:Set Product

Definition

Let $S$ and $T$ be sets.

Let $P$ be a set.

Let $\phi_1: P \to S$ and $\phi_2: P \to T$ be mappings such that:

For all sets $X$ and all mappings $f_1: X \to S$ and $f_2: X \to T$ there exists a unique mapping $h: X \to P$ such that:

$\phi_1 \circ h = f_1$

$\phi_2 \circ h = f_2$

that is, such that:

$\begin{xy}\xymatrix@+1em@L+3px{ & X \ar[ld]_*+{f_1} \ar@{-->}[d]^*+{h} \ar[rd]^*+{f_2} \\ S & P \ar[l]^*+{\phi_1} \ar[r]_*+{\phi_2} & T }\end{xy}$

Then the ordered triple $\struct {P, \phi_1, \phi_2}$ is called a product of $S$ and $T$.

Let $\family {S_i}_{i \mathop \in I}$ be an indexed family of sets.

Let $P$ be a set.

Let $\family {\phi_i}_{i \mathop \in I}$ 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 $\family {f_i}_{i \mathop \in I}$ 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 $i \in I$:

$\quad \quad \begin {xy} \xymatrix@+1em@L+3px { X \ar@{-->}[d]_*+{h} \ar[dr]^*+{f_i} \\ P \ar[r]_*{\phi_i} & S_i } \end {xy}$

Then the ordered pair $\struct {P, \family {\phi_i}_{i \mathop \in I} }$ is called a product of (the family) $\family {S_i}_{i \mathop \in I}$.

The mappings $\phi_i$ are the projections of $P$.