boundary of $M \times I$ where $M$ is the Möbius band

Question

Let $M$ be the Möbius band and $I$ be the closed interval $[0,1]$. What is the boundary of $M \times I$? Is it orientable?

What can I do when I want to know the boundary of such space? Please give an advice.

Thank you for your help!

Answer 1

Unless you mean a Möbius band that's modeled on an open interval $(-1,1)$ rather than the usual closed interval $[-1,1]$, this will be (in the smooth world) a manifold with corners, not a manifold with boundary; in the topological world, no problem.

In general, $\partial(M\times N) = \partial M\times N \cup M\times\partial N$. So $$\partial (M\times I) \approx S^1\times I \cup M\times \partial I \approx S^1\times I \cup M\times\{0,1\}.$$ The latter factor is certainly not orientable.

Answer 2

$M \times [0,1]$ is homeomorphic to the "solid Klein bottle", and its boundary is the ordinary 2-dimensional Klein bottle, which is nonorientable.