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?
Thanks in advance!