A single point is a one-dimensional polytope, and is entirely of content, without boundary. The problem comes when you try to divide space with an equal-sign (ie upper and lower half, or $x \gt a$ vs $x \lt a$), when dividing a point is meaningless. Therefore a point exists in an undividable space, and since a boundary is a division, a point cannot have a boundary. The definition ought imply that $k>0$.
Another is by tomasz:
If you have a manifold with boundary and take a submanifold, for well-behavedeness purposes you sometimes want the boundary of the submanifold to be its intersection with boundary of the larger manifold. In this case, if the submanifold is a singleton, the single point might or might not be in the boundary.
So I am wondering if I can ask for some clarification here? Thank you.