Topological boundary as a submanifold

Question

Let $U$ be an open subset of a smooth $n$-manifold. Consider $\partial U$ the topological boundary of $U$.

Is the following true ? :

If $\partial U$ is a smooth $n-1$ submanifold without boundary, then $\overline U$ is a manifold with boundary (and the boundary of $\overline U$ as a manifold with boundary is $\partial U$).

Answer 1

This is false. Consider e.g. $S^2$ with the equator $E \sim S^1$, with $U = S^2 \setminus E$. Then $E$ is indeed a smooth 1-dim submanifold of $S^2$, but $\bar U = S^2$ does not have a boundary.