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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}$(\alpha)$.

\s\s
Defn:  Let $\{A_\alpha\}_{\alpha \in J}$ be an indexed family of sets. 
\break 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 $\alpha \in J$.
\s

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

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

\s
\centerline{$\{\Pi_{\alpha
\in J} {U_\alpha} ~|~ U_\alpha$ open in $X_\alpha\}$.}
\s
\s
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 ${\cal S} = \cup_{\alpha \in J}
{\cal S}_\alpha$.

\eject
Thm 19.1, 2: Comparison of box and product topologies. Let ${\cal 
B}_\alpha$ be a basis for $X_\alpha$

Basis for the box topology:  $\{\Pi U_\alpha ~|~  U_\alpha$ open in 
$X_\alpha \}$

\centerline{or $\{\Pi B_\alpha ~|~  B_\alpha \in {\cal B}_\alpha \}$}


Basis for the product topology:  

$\{\Pi U_\alpha ~|~  U_\alpha$ open in $X_\alpha$, 

\rightline{$U_\alpha = 
X_\alpha$ for all but finitely many $\alpha \}$}

or $\{\Pi B_\alpha ~|~  B_{\alpha_i} \in {\cal B}_{\alpha_i}, ~i = 1, ..., 
n,$ 

\rightline{$B_\alpha = X_\alpha$ 
for $\alpha \not= {\alpha_i}, ~i = 1, ..., n \}$}


Hence box topology is finer then the product topology

Thm 19.3:  Let $A_\alpha$ be a subspace of $X_\alpha$.  Then $\Pi 
A_\alpha$ is a subspace of $\Pi
X_\alpha$ if both products are given the box topology or if both products 
are given the product
topology.

Thm 19.4:  If $X_\alpha$ is Hausdorff for all $\alpha$ then $\Pi X_\alpha$ 
is Hausdorff in both the box and product topologies.


HW p. 118: 3, 5, 6, 7


Thm 19.5:  $\Pi \overline{A_\alpha} = \overline{\Pi A_\alpha}$ in both the 
box and product topologies.

Thm 19.6:  Suppose $f_\alpha: X \rightarrow Y_\alpha$.  Define $f: X 
\rightarrow \Pi_{\alpha \in A}
Y_\alpha$ by $f(x) = (f_\alpha(x))_{\alpha \in A}$.  Let $\Pi X_\alpha$ 
have the product topology.  
Then $f$ is continuous is and only if $f_\alpha$ is continuous $\forall$ 
$\alpha$





\end

\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: