Activity 20.2.
Let \((X, \tau)\) and \((Y, \tau_Y)\) be topological spaces, and let \(\CB\) be as defined in (20.1). Prove that \(\CB\) is a basis for a topology on \(X \times Y\text{.}\)
Section The Topology on a Product of Topological Spaces
In our preview activity we learned that we cannot make a topology on a product \(X \times Y\) of topological spaces \((X, \tau_X)\) and \((Y , \tau_Y)\) with just the sets of the form \(U \times V\) where \(U \in \tau_X\) and \(V \in \tau_Y\) as the open sets since the collection of these sets is not closed under arbitrary unions. What we can do instead is consider these unions of all of the sets of the form \(U \times V\text{,}\) where \(U\) is open in \(X\) and \(V\) is open in \(Y\text{.}\) In other words, consider these sets to be a basis for the topology on \(X \times Y\text{.}\)
The argument from Activity 20.2 can be extended to a product of any finite number of topological spaces. Let \(n\) be a positive integer and let \((X_i, \tau_i)\) be topological spaces for \(i\) from \(1\) to \(n\text{.}\) Let
\begin{equation*}
\CB = \left\{ \Pi_{i=1}^n O_i \mid O_i \text{ is open in } X_i\right\}\text{.}
\end{equation*}
Since \(X_i \in \tau_i\) for every \(i\text{,}\) every point in \(\Pi_{i=1}^n X_i\) is in a set in \(\CB\text{.}\) So \(\CB\) satisfies condition 1 of a basis. Now we show that \(\CB\) satisfies the second condition of a basis. Let \(B_1 = \Pi_{i=1}^n U_i\) and \(B_2 = \Pi_{i=1}^n V_i\) for some open sets \(U_i\text{,}\) \(V_i\) in \(X_i\text{.}\) Suppose \((x_i) \in (B_1 \cap B_2)\text{.}\) Then for each \(j\) we have \(x_j \in U_j \cap V_j\) and so
\begin{equation*}
(x_i) \in \Pi_{i=1}^n (U_i \cap V_i)\text{.}
\end{equation*}
Since \(U_i \cap V_i\) is an open set in \(X_i\text{,}\) it follows that \(\Pi_{i=1}^n (U_i \cap V_i)\) is in \(\CB\text{.}\) Thus, \(\CB\) is a basis for a topology on \(X \times Y\text{.}\)
This topology generated by products of open sets is called the box or product topology.
Definition 20.1.
Let \((X_{\alpha}, \tau_{\alpha})\) be a collection of topological spaces for \(\alpha\) in some finite indexing set \(I\text{.}\) The box topology or product topology on the product \(\Pi_{\alpha \in I} X_{\alpha}\) is the topology with basis
\begin{equation*}
\CB = \left\{ \Pi_{\alpha \in I} U_{\alpha} \mid U_{\alpha} \in \tau_{\alpha} \text{ for each } \alpha \in I \right\}\text{.}
\end{equation*}
So we can always make the product of topological spaces into a topological space using the box topology.
