Prove a property of covering maps: p × p′

Nov 2012
I just read a property of covering maps:
If p : EB and p′ : E′ → B′ are covering maps, then so is the map p × p′ : E × E′ → B × B′ given by (p × p′)(e, e′) = (p(e), p′(e′)).

But how do I prove it?

Since I'm new to this area, so I first thought of starting with the definition. But then given an open set U of B × B′, I can't break it with regard to B and B'. What should I do?
Jul 2005
My house.
Αn open set in \(\displaystyle B\times B'\) will be of the form \(\displaystyle U\times U'\), and so you can use the covering property component-wise:
\(\displaystyle (p\times p')^{-1}(U\times U')=(p^{-1}(U),p'^{-1}(U'))\).