Let be $M$ a smooth manifold with boundary and $N \subset M$ a submanifold with boundary, such $N$ is a closed subset of $M$ in the topological sense. Denote $\partial N$ be the boundary of $N$ as a manifold, and $\mathring{N}=\overline{N}-Int(N)$ the boundary of $N$ in the topological sense. Is it true
$\mathring{N}=N$ if $dim(N)<dim(M)$ $\hspace{5 mm}$ and $\hspace{5 mm}$ $\mathring{N}=\partial N$ if $dim(N)=dim(M)$ ?
Does the result change if $N$ and $M$ are non smooth, but $C^k$ manifolds with some $k\in \mathbb{N}_0 $?