Topology: An Inquiry-Based Approach

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 $X\times Y\text{.}$

Since $U_i\cap V_i$ is an open set in $X_i\text{,}$ it follows that ${\mathrm{\Pi }}_{i=1}^n(U_i\cap V_i)$ is in $\mathcal{B}\text{.}$ Thus, $\mathcal{B}$ 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.

So we can always make the product of topological spaces into a topological space using the box topology.