Prove that for any topological polyhedra, $X$, $Y$, the product $X \times Y$ has the Euler characteristic $\chi(X \times Y)=\chi(X)\chi(Y)$
I know that for polyhedron $P$ which is homemorphic to a closed surface $S$. If $P$ has $f $ faces, $ e$ edges and $ v$ vertices. Then $\chi(P)=f-e+v $.
I believe that taking the product, still makes the shape a polyhedra, but I am not sure how to count the number of edges, vertices and faces