The description of the proposition is here.And here is the proof of showing $M \cup_h N$ is locally Euclidean of dimension $n$:
I‘m encountering difficulty in applying the gluing lemma to $\Phi^{-1}$. The Gluing Lemma, as stated in the book, is as follows:
Let $X$ and $Y$ be topological spaces, and let $\left\{A_i\right\}$ be either an arbitrary open cover of $X$ or a finite closed cover of $X$. Suppose that we are given continuous maps $f_i: A_i \rightarrow Y$ that agree on overlaps: $\left.f_i\right|_{A_i \cap A_j}=$ $\left.f_j\right|_{A_i \cap A_j}$. Then there exists a unique continuous map $f: X \rightarrow Y$ whose restriction to each $A_i$ is equal to $f_i$.
However, I am struggling with the application of this lemma when the sets defined by $y_n \ge 0$ or $y_n \le 0$ do not seem to be either open or closed in $\mathbb{R}^n$. I wonder if I am overlooking some crucial points that would allow me to apply the lemma in this context.
Could anyone provide insights or point out what I might be missing here?
My second question is that how can we know that the image of $\Phi$ is an open subset of $\mathbb{R^n}$?