As an independent project and after reading the first three chapters in the book $\textit{Differential Topology}$ by Victor Guillemin and Alan Pollack, I am trying to prove the Poincare-Hopf Theorem. So, after analyzing some of the conditions given in the theorem and the definition of manifold given in the book I ran into a question. The theorem says:
If $F$ is a smooth vector field on the compact, oriented $k$-dimensional manifold $X \subset \mathbb{R}^n$ with only finitely many zeroes, then the global sum of the indices of $F$ equals the Euler characteristic of $X$.
So, is $X$ a submanifold of $\mathbb{R}^n$? Is this true for the theorem? Along the book I think the author did not mention anything about all what is implied for a general manifold (second countable, and all that).
Maybe this is a basic question in differential topology but I would really appreciate your help here, since this could drastically change my approach.
Thanks for all!!