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).

**Clarification**

The actual situation where this arose for me was the following: I have an open ball $B$ in euclidean space and a submanifold $N$ of $B$ as discussed above. I want to be able to say that for (almost*) every closed ball $B'\subset B$ that $N\cap B'$ is a submanifold with boundary in the usual well understood sense AND $\partial (B'\cap N) \subset \partial B'$. To ensure this one would like to say that $\partial N\subset \partial B$. This of course raises legitimate questions as to what $\partial N$ means. It seems a reasonable definition for my purposes is $\partial N=\bar{N}\backslash N$ (here $\bar{N}$ is the closure in the ambient space). However, I worry there are subtle problems with such a definition and was wondering if there was a canonical definition (which perhaps there is not).

*One can't expect to have this for every closed ball due to a failure of transverality between $\partial B'$ and $N$.