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19. The Product Topology.

Defn:  Let $J$ be an index set.  Given a set $X$, a \break
{\bf J-tuple} of
elements of $X$ is a function ${\bf x}: J \rightarrow X$.  The {\bf
$\alpha$th coordinate of x} = $x_\alpha$ = {\bf x}$(\alpha)$.


Defn:  Let $\{A_\alpha\}_{\alpha \in J}$ be an indexed family of sets.  Let
$X = \cup_{\alpha \in J}A_\alpha$.  The {\bf cartesian product} of
$\{A_\alpha\}_{\alpha \in J}$, denoted by $\Pi_{\alpha \in J} {A_\alpha}$,
is defined to the the set of all J-tuples $(x_\alpha)_{\alpha \in J}$ of
elements of $X$ such that $x_\alpha \in A_\alpha$ for each $\alpha \in J$.

That is, it is the set of all functions \break
${\bf x}: J \rightarrow
\cup_{\alpha \in J}A_\alpha$ such that ${\bf x}(\alpha) \in A_\alpha
\forall \alpha \in J$.

Defn:  The {\bf box topology} on $\Pi_{\alpha \in J} {X_\alpha}$ is the
topology
generated by the basis 

\centerline{$\{\Pi_{\alpha
\in J} {U_\alpha} ~|~ U_\alpha$ open in $X_\alpha\}$.}

Defn:  Let ${\cal S}_\alpha = \{ \pi_\alpha^{-1}(U) ~|~ U$ open in
$X_\alpha\}$ \hfil \break
The {\bf product topology} on $\Pi_{\alpha \in J} {X_\alpha}$ is the
topology 
generated by the subbasis ${\cal S} = \cup_{\alpha \in J}
{\cal S}_\alpha$.




\end

Let $\pi_j: \Pi_{i=0}^n X_i \rightarrow X_j$, $\pi_j(x_1, ..., x_n) = x_j$.
$\pi_j$ is the projection of $\Pi_{i=0}^n X_i $ onto the $j$th component.


Note: