This question has been bothering me for a while now and I fear is a result of a gap in my education.

Just so we are all on the same page I'll define submanifold... Suppose we have a subset $N\subset M$ where $M$ is a $n$-manifold. We say that $N$ is a $k$-dimensional submanifold if for each point $p\in N$ there is a neighborhood $U$ of $p$ in $M$ so that the component of $U\cap N$ containing $p$ may be parameterized by a $C^\infty$ map of an open subset of $\mathbb{R}^k$. For the sake of concreteness (as I think it might matter) let us suppose $M$ has a metric and is complete with respect to this metric.

My question is: Is there a correct way to talk about $\partial N$ the boundary of $N$ or does one need more structure? For instance if $N$ has the structure of an integer rectifiable current then the boundary is defined unambiguously.

To clarify I bit what I'm worried about I'll give some examples:

Suppose one starts with a closed submanifold $N$ and deletes a finite number of points what is the boundary of new submanifold? What if you instead delete a Cantor set?

$N$ is non-proper. For instance if the induced metric on $N$ is complete but $N$ lies entirely within a compact region of $M$ (so $N$ has points of accumulation in $M$). Does $N$ not have a boundary? (This would I believe, prevent $N$ from being a current).