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 $\{X_{i}\}_{\i\in I}$ .

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

	$\displaystyle\prod_{i\in I}f_{i}:\prod_{i\in I}A_{i}\longrightarrow\prod_{i\in I% }B_{i}$
	$\displaystyle\Big{(}\prod_{i\in I}f_{i}\Big{)}(a_{i})_{i\in I}:=(f_{i}(a_{i}))% _{i\in I}$

0.1 Properties:

•

If $f_{i}:A_{i}\longrightarrow B_{i}$ and $g_{i}:B_{i}\longrightarrow C_{i}$ 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.

Title	product map
Canonical name	ProductMap
Date of creation	2013-03-22 17:48:28
Last modified on	2013-03-22 17:48:28
Owner	asteroid (17536)
Last modified by	asteroid (17536)
Numerical id	6
Author	asteroid (17536)
Entry type	Definition
Classification	msc 03E20