Subspace not open of a differentiable manifold

Question

Suppose $M$ is an orientable differentiable manifold with dimension $n$.

$U$ is a subspace of $M$. If $U$ is not open, is it true that $U$ also is an orientable differentiable manifold ? I need a possible counterexample.

Answer 1

"Not open" is a pretty weak condition! Almost none of the subsets of $M$ are open, and almost none are submanifolds. You could take, for instance, $M = \mathbb{R}^2$ and let $U$ be the union of the $x$ and $y$ axes. That's not open in $M$, and it's not a manifold at all.

The fact that you've asked this probably means that you don't understand what "open" means or that you don't understand what "manifold" means, or probably both. I'd recommend going over those again.