PlanetMath: product map

(more info)

Math for the people, by the people.

Main Menu

sections Encyclopædia
Papers
Books
Expositions

meta Requests (231)
Orphanage
Unclass'd
Unproven (514)
Corrections (23)
Classification

talkback Polls
Forums
Feedback
Bug Reports

downloads Snapshots
PM Book

information News
Docs
Wiki
ChangeLog
TODO List
Legalese
About

Entry average rating: No information on entry rating

product map

(Definition)

Notation: If $\{X_i\}_{i \in I}$ is a collection of sets (indexed by ) 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 and $f_i:A_i\longrightarrow B_i$ a collection of functions.

The product map is the function

$\displaystyle \prod_{i \in I} f_i : \prod_{i \in I} A_i \longrightarrow \prod_{i \in I} B_i$
$\displaystyle \Big( \prod_{i \in I} f_i \Big) (a_i)_{i \in I} := (f_i(a_i))_{i \in I}$

Properties:

If $f_i:A_i\longrightarrow B_i$ and $g_i:B_i\longrightarrow C_i$ are collections of functions then
$\displaystyle \prod_{i \in I} g_i \circ \prod_{i \in I} f_i = \prod_{i \in I} g_i \circ f_i$
$\displaystyle \prod_{i \in I} f_i$ is injective if and only if each is injective.
$\displaystyle \prod_{i \in I} f_i$ is surjective if and only if each 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 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 is a group (ring or algebra) homomorphism.