Notation: If $\{X_i\}_{i \in I}$ is a collection of sets (indexed by $I$ ) then $\displaystyle \prod_{i \in I} X_i$ denotes the generalized Cartesian product of $\{X_i\}_{\i \in I}$ .

Let $\{A_i\}_{i\in I}$ and $\{B_i\}_{i\in I}$ be collections of sets indexed by the same set $I$ and $f_i:A_i\longrightarrow B_i$ a collection of functions.

The product map is the function

Properties:

• If $f_i:A_i\longrightarrow B_i$ and $g_i:B_i\longrightarrow C_i$ are collections of functions then
  
• $\displaystyle \prod_{i \in I} f_i$ is injective if and only if each $f_i$ is injective.
• $\displaystyle \prod_{i \in I} f_i$ is surjective if and only if each $f_i$ is surjective.
• Suppose $\{A_i\}_{i \in I}$ and $\{B_i\}_{i \in I}$ are topological spaces. Then $\displaystyle \prod_{i \in I} f_i$ is continuous (in the product topology) if and only if each $f_i$ is continuous.
• Suppose $\{A_i\}_{i \in I}$ and $\{B_i\}_{i \in I}$ are groups, or rings or algebras. Then $\displaystyle \prod_{i \in I} f_i$ is a group (ring or algebra) homomorphism if and only if each $f_i$ is a group (ring or algebra) homomorphism.