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 \(\R^2\) 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.
Preview Activity20.1.
Let \(X = \{a,b,c\}\) with \(\tau_X = \{\emptyset, \{a\}, \{b\}, \{a,b\}, \{a,c\}, X\}\text{,}\) and let \(Y = \{1,2\}\) with \(\tau_Y = \{\emptyset, \{1\}, Y\}\text{.}\)
(a)
Let
\begin{equation}
\CB = \{U \times V \mid U \in \tau_X \text{ and } V \in \tau_Y\}\text{.}\tag{20.1}
\end{equation}
List all of the sets in \(\CB\) along with their elements.
(b)
Assume that all of the sets in \(\CB\) are open sets in \(X \times Y\text{.}\) Should the set \(A = \{(a,1), (a,2), (b,1)\}\) be an open set in \(X \times Y\text{?}\) Is the set \(A\) of the form \(U \times V\) for some open sets \(U\) in \(X\) and \(V\) in \(Y\text{?}\) Explain. Is \(\CB\) a topology on \(X \times Y\text{?}\)
(c)
If \(\CB\) is not a topology on \(X \times Y\text{,}\) what is the smallest collection of sets would we need to add to \(\CB\) to make a topology on \(X \times Y\text{?}\) Explain your process.
