Let \(f: A \to B\) and \(g: C \to D\) be continuous functions. Let us define a map \(f \times g: A \times C \to B \times D\) by the equation
\[ (f \times g)(a \times c) = f(a) \times g(c). \]
Show that \(f \times g\) is continuous.
Solution
A basis for \(B\times D\) consists of sets \(U\times V\) with \(U\subseteq B\) open and \(V\subseteq D\) open. For such a basic set, \[ (f\times g)^{-1}(U\times V)=f^{-1}(U)\times g^{-1}(V). \] Since \(f\) and \(g\) are continuous, the two factors on the right are open in \(A\) and \(C\), so their product is open in \(A\times C\). Thus inverse images of basis elements are open, hence \(f\times g\) is continuous.