# Independence of definition of boundary point of a manifold

I'm using the book Differential Forms and Applications by Do Carmo in order to understand the theorem of Stokes on compact manifolds and I'm stuck in the following lemma:

My doubt is why $(f_1^{-1} \circ f_2)(V)$ has points $(x_1,\cdots,x_n)$ such that $x_1 >0$? The image suggests that $(f_1^{-1} \circ f_2)(V)$ is a neighborhood of $q$ has points $(x_1,\cdots,x_n)$ such that $x_1 >0$, but $\text{Im} (f_1^{-1} \circ f_2)(V) \subset f_1^{-1}(W) \subset U_1 \subset H^n$, then I should not consider a neighborhood of $q$ according to the topology of $H^n$? If I can see the neighborhood $(f_1^{-1} \circ f_2)(V)$ of $q$ according to the topology of $\mathbb{R}^n$, why can I see of this form?

• The inverse function theorem applies to functions from (an open set of) $\mathbb{R}^n$ to $\mathbb{R^n}$, and when its conditions are satisfied it gives the existence of an inverse defined on an open subset of $\mathbb{R}^n$, not an open set of some other space. It is being applied to $(f_1^{-1}\circ f_2):U\subset\mathbb{R}^n\to \mathbb{R}^n$. It implies the existence of an inverse $g$ of $f_1^{-1}\circ f_2$ restricting its domain and image to open sets of $\mathbb{R}^n$. A neighborhood of $q$ in $\mathbb{R}^n$ contains points with $x_1<0$. – user569098 Jun 27 '18 at 21:47