Let $(X, \tau_1)$, $(Y, \tau_1)$ be topological spaces with $\# (X) = \text{number of connected components} < \infty$ and $\# (Y) = \text{number of connected components} < \infty$.
How can I show that this implies $\# (X \times Y) = \# (X) \cdot \# (Y)$ (in the sense of the product topology)?