product map

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 ${\{X_i\}}_{ı\in I}$.

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

The product map is the function

${\displaystyle \prod _{i\in I}}{f}_i:{\displaystyle \prod _{i\in I}}{A}_i⟶{\displaystyle \prod _{i\in I}}{B}_i$

$\left({\displaystyle \prod _{i\in I}}{f}_i\right){(a_i)}_{i\in I}:={(f_i(a_i))}_{i\in I}$

0.1 Properties:

• If $f_i:A_i⟶B_i$ and $g_i:B_i⟶C_i$ are collections of functions then

  $$\prod _{i\in I}{g}_i\circ \prod _{i\in I}{f}_i=\prod _{i\in I}{g}_i\circ f_i$$

• ${\displaystyle \prod _{i\in I}}{f}_i$ 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 (http://planetmath.org/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 ) homomorphism if and only if each $f_i$ is a group (ring or ) homomorphism.