So I'm reading Allan Pollack and Victor Guillemin's Differential Topology. We haven't so far defined compact submanifolds explicitly but I'm guessing its just a submanifold (which is defined as a subset of manifold such that the subset itself is a manifold) which is closed and bounded.
I don't understand how compact submanifolds are manifolds. Since they're compact, the set contains its boundary. And we know that for any point in the boundary, any neighbourhood around it will have points not in the manifold. Thus, we can't form a neighbourhood around a point in the manifold which implies it fails the definition of a manifold.
So how should I be interpreting the phrase: compact submanifold ?
The book, so far, has not defined manifolds with boundary, and I'm not exactly sure what those are either way. This word came up in the following context:
Generalization of the inverse function theorem: Let $f:X\to Y$ be a smooth map that is one-to-one on a compact submanifold $Z$ of $X$. Suppose that $x\in Z$, $$df_x: T_x(X) \to T_{f(x)}(Y)$$ is an isomorphism. Then $f$ maps $Z$ diffeomorphically onto $f(Z)$.
I think you realized it already, but there is a difference between topological boundary and manifold boundary. Topological boundary is the one in which a set assumes as a subset of some ambient space, whereas the manifold boundary is even more general because the set itself is the whole manifold say $M$ i.e it is not assume to be a subset of some larger space.