Solution: Let $X$ and $Y$ be two Hausdorff topological spaces. We need to show that $X \times Y$ is Hausdorff. Let $\left( x_1, y_1 \right) $ and $\left( x_2, y_2 \right) $ be two distinct points in $X \times Y$. If $x_1 \neq x_2$, then using $X$ is Hausdorff, there exist two disjoint open sets in $X$ say, $U_X$ and $V_X$ such that $x_1 \in U_X$ and $x_2 \in V_X$. Consider the open sets $U_X \times Y$ and $V_X \times Y$. It is clear that \[ \left( x_1, y_1 \right) \in U_X \times Y \text{ and } \left( x_2, y_2 \right) \in V_X \times Y. \] Also, as $U_X$ and $V_X$ are disjoint, their product with $Y$ will also be disjoint.