Is the product of covering maps a covering map?

Question

I have a question about covering maps. If $\phi_1: X_1 \rightarrow Y_1$ is a covering map, and $\phi_2: X_2 \rightarrow Y_2$ is a covering map, then is it true that $\phi_1 \times \phi_2: X_1 \times X_2 \rightarrow Y_1 \times Y_2$ a covering map?

Answer 1

Yes. This is a case where you should just follow the definition. The map $\phi_1\times\phi_2$ is continuous and surjective, and every point of $Y_1\times Y_2$ has a neighborhood that is evenly covered (remember the definition of the product topology).