Boundary of a manifold

Question

The product of manifolds without boundaries is always a manifold without boundary?

$\textbf{Example:}$ $S^2\times \mathbb R$ has a boundary?

Answer 1

Yes. More generally if $M$ and $N$ are boundary manifolds, then $M\times N$ is a manifold with boundary and $$\partial (M\times N)=\partial M\times N\cup M\times \partial N.$$ So if $\partial M=\partial N=\emptyset$ then $\partial (M\times N)=\emptyset$. Note that here I am talking about topological manifolds.