# The boundary is disjoint from the interior in 2d manifold

I'm self-studying on Lee's Introduction to Topological Manifolds, and I'm stuck on a problem (8-6). I'm missing something really basic, I fear. I must prove that the set of the boundary points of a 2-dimensional manifold with boundary is disjoint from the set of interior points. Then show that a 2-manifold is a manifold iff its boundary is empty.

Now, my reasoning is the following. The boundary of a 2d manifold with boundary $$M$$ must be a 1d manifold. If the manifold is compact, it must be homeomorphic to $$S^1$$, and $$\pi_1(\partial M,q) \simeq \pi_1(S^1,1)$$ for every $$q$$.

Given the 2d classification, $$Int M$$ is homeomorphic to $$S^2$$, the connected sum of tori, or the connected sum of projected planes.

In the case of the sphere, if $$\exists q \in IntM \cap \partial M$$, I can construct retractions, by using path connection of the sphere, and show that $$\pi_1(S^1,1)$$ must be isomorphic to the trivial fundamental group, which is absurd, since it is infinite cyclic.

For the other cases, I cannot see how to proceed... And what about non-compact manifolds? I would loose the classification theorem (or at least the simple one).

Thank you