Cartesian product of compact triangulated spaces

Question

Let $X$ and $Y$ two compact triangulated spaces, I am trying to show that $X\times Y$ is also a compact (this is obvious) triangulated space and $$\chi(X\times Y)=\chi(X)\cdot\chi(Y)$$

Any tips on where to start?

Answer 1

Let $A$ be a simplex in $X$ with vertices $a_0,a_1,...,a_n$ and let $B$ be an simplex in $Y$ with vertices $b_0,b_1,...,b_m$. Consider the sequence with $(n+m+1)$ elements of pairs $(a_i,b_j)$ s.t in them each $(a_i,b_j)$ is followed either by $(a_{i+1},b_j)$ or $(a_i,b_{j+1})$. So this set is defined the sequence of vertices of an $(m+n)$ simplex in $X\times Y$. Now if we vary $A, B$ over all set of simplices in $X, Y$ and construct new simplices of $X\times Y$, then this set of simplices together with their faces form a triangulation of $X\times Y$. Now from here it is very straight forward to count how many simplices of each dimension are there in $X\times Y$ ( depending on the triangulation of $X$ and $Y$). And this gives you the result that $\chi (X\times Y)= \chi(X) \times \chi(Y)$.