Topology: An Inquiry-Based Approach

In Chapter 11 we saw how we can make a Cartesian product of two metric spaces into a metric space. This is exactly the construction that allows us to work with the Cartesian plane ${\mathbb{R}}^2$ as a metric space with the usual metric. As we discussed in Chapter 12, every metric space is a topological space, but not every topological space is metrizable. So knowing how to make a product of metric spaces into a metric space still leaves open the question of how we can make the product of topological spaces into a topological space. If we have two topological spaces $(X,{\tau }_X)$ and $(Y,{\tau }_Y)\text{,}$ a natural approach to this problem might be to take as the open sets in $X\times Y$ the sets of the form $U\times V$ where $U\in {\tau }_X$ and $V\in {\tau }_Y\text{.}$ We investigate this idea in Preview Activity 20.1.