While I am reading "Introduction to Topological Manifolds" by John M. Lee, I come to see the following paragraph in the proof of Theorem 5.10 pp. 102.
Note that Int$\ e\cap\ $Int$\ e'$ is open in Int$\ e$. On the other hand, $e'$ is a compact subset of the Hausdorff space $M$, so it is closed in $M$, and therefore Int$\ e\cap\ $Int$\ e'=$Int$\ e\cap e'$ is closed in Int $e$.
I don't understand why Int$\ e\cap\ $Int$\ e'=$Int$\ e\cap e'$.
Question 1: Is this because no vertex of $G_n$ lies in the interior of any edge of $S$?
Question 2: Can I consider $G_n$ as the finite union of closed intervals in $\mathbb{R}$ as we are talking about 1-simplex?
Thanks for any help.