product map
Notation: If ${\{{X}_{i}\}}_{i\in I}$ is a collection^{} of sets (indexed by $I$) then $\prod _{i\in I}}{X}_{i$ denotes the generalized Cartesian product of ${\{{X}_{i}\}}_{\u0131\in I}$.
Let ${\{{A}_{i}\}}_{i\in I}$ and ${\{{B}_{i}\}}_{i\in I}$ be collections of sets indexed by the same set $I$ and ${f}_{i}:{A}_{i}\u27f6{B}_{i}$ a collection of functions.
The product map is the function
$\prod _{i\in I}}{f}_{i}:{\displaystyle \prod _{i\in I}}{A}_{i}\u27f6{\displaystyle \prod _{i\in I}}{B}_{i$  
$\left({\displaystyle \prod _{i\in I}}{f}_{i}\right){({a}_{i})}_{i\in I}:={({f}_{i}({a}_{i}))}_{i\in I}$ 
0.1 Properties:

•
If ${f}_{i}:{A}_{i}\u27f6{B}_{i}$ and ${g}_{i}:{B}_{i}\u27f6{C}_{i}$ are collections of functions then
$$\prod _{i\in I}{g}_{i}\circ \prod _{i\in I}{f}_{i}=\prod _{i\in I}{g}_{i}\circ {f}_{i}$$ 
•
$\prod _{i\in I}}{f}_{i$ is injective^{} if and only if each ${f}_{i}$ is injective.

•
$\prod _{i\in I}}{f}_{i$ is surjective^{} if and only if each ${f}_{i}$ is surjective.

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Suppose ${\{{A}_{i}\}}_{i\in I}$ and ${\{{B}_{i}\}}_{i\in I}$ are topological spaces^{}. Then $\prod _{i\in I}}{f}_{i$ is continuous^{} (http://planetmath.org/Continuous) (in the product topology) if and only if each ${f}_{i}$ is continuous.

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Suppose ${\{{A}_{i}\}}_{i\in I}$ and ${\{{B}_{i}\}}_{i\in I}$ are groups, or rings or algebras. Then $\prod _{i\in I}}{f}_{i$ 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  20130322 17:48:28 
Last modified on  20130322 17:48:28 
Owner  asteroid (17536) 
Last modified by  asteroid (17536) 
Numerical id  6 
Author  asteroid (17536) 
Entry type  Definition 
Classification  msc 03E20 