In Guillemin and Pollack they state:
Proposition. The product of a manifold without boundary, $X$ and a manifold with boundary $Y$ is another manifold with boundary. Furthermore:
$$ \partial(X \times Y) = X \times \partial Y$$
and:
$$ \dim ( X \times Y) = \dim(X) + \dim(Y)$$
The proof goes as follows:
If $U \in R^k$ and $V \in H^l$ are open, then:
$$ U \times V \in R^k \times H^l = H^{k+l} $$
is open. Moreover, if $\phi : U \to X$ and $\psi : V \to Y$ are local parameterizations, so is $\phi \times \psi : U \times V \to X \times Y$.
So, I see that as any local parameterization is in $R^k \times H^l = H^{k+l}$ this implies the $\dim$ is $k + l$, and therefore $\dim(X) + \dim(Y)$, but it is not clear to me why this proof illustrates:
$$\partial (X \times Y) = X \times \partial Y$$
Could someone clarify why $\partial (X \times Y) = X \times \partial Y$? If you need more clarification, please ask.