\magnification 2100 %%\magnification 2400 \parindent 0pt \parskip 13pt \pageno=18 \hoffset -0.2truein \hsize 7.1truein \vsize 9.2truein \def\u{\vskip -10pt} \def\v{\vskip -5pt} \def\s{\vskip 5pt} \parindent 0pt \parskip 8pt \nopagenumbers \hsize 7.5truein \vsize 9.9 truein \hoffset -0.45truein \def\u{\vskip -7pt} \def\v{\vskip -5pt} %%\input ../../../PAPERS/GOLD/psfig 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: