 Solution

The product topology has as a basis of a topology all sets $U\times V$, where $U$ is open in $X$ and $V$ is open in $Y$. Open sets are arbitrary unions of these rectangles. The projection maps are $p(x,y)=x$ and $q(x,y)=y$. For any open $U\subset X$ and $V\subset Y$,These are open rectangles, including the empty case. Taking inverse images of open sets therefore proves that both projection maps are continuous functions.