product map
ProductMap
2008-02-14 14:30:24
2008-02-14 14:36:20
Definition
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{\bf 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 {\bf product map} is the function
\begin{align*}
\prod_{i \in I} f_i : \prod_{i \in I} A_i \longrightarrow \prod_{i \in I} B_i\\
\Big( \prod_{i \in I} f_i \Big) (a_i)_{i \in I} := (f_i(a_i))_{i \in I}
\end{align*}
\subsection{Properties:}
\begin{itemize}
\item If $f_i:A_i\longrightarrow B_i$ and $g_i:B_i\longrightarrow C_i$ are collections of functions then
\begin{displaymath}
\prod_{i \in I} g_i \circ \prod_{i \in I} f_i = \prod_{i \in I} g_i \circ f_i
\end{displaymath}
\item $\displaystyle \prod_{i \in I} f_i$ is injective if and only if each $f_i$ is injective.
\item $\displaystyle \prod_{i \in I} f_i$ is surjective if and only if each $f_i$ is surjective.
\item Suppose $\{A_i\}_{i \in I}$ and $\{B_i\}_{i \in I}$ are topological spaces. Then $\displaystyle \prod_{i \in I} f_i$ is \PMlinkname{continuous}{Continuous} (in the product topology) if and only if each $f_i$ is continuous.
\item 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 \PMlinkescapetext{algebra}) homomorphism if and only if each $f_i$ is a group (ring or \PMlinkescapetext{algebra}) homomorphism.
\end{itemize}