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I'm confused about the definition of an interior chart adopted by John M. Lee. The following is a snapshot of the text, and the part that bothers me is underlined with red. Please tell me why an open subset of $\mathbb{H}^n$ that does not intersect $\partial\mathbb{H}^n$ can be seen as an open set in the Euclidean space. I have a hard time using the subspace topology to conclude this result.
If $U$ is an open set of $\mathbb H^n$, then there exists an open set $V$ of $\mathbb R^n$ such that $U = V \cap \mathbb H^n$. Now, clearly $\mathbb H^n = \partial \mathbb H^n \cup \operatorname{Int} \mathbb H^n$, which means that $$U = (V \cap \partial \mathbb H^n) \cup (V \cap \operatorname{Int} \mathbb H^n).$$ If $U \cap \partial \mathbb H^n = \varnothing$, then $V \cap \partial \mathbb H^n = \varnothing$ and that implies that $U = V \cap \operatorname{Int} \mathbb H^n$, that is, $U$ is an open set in $\mathbb R^n$ since it is a intersection of two open sets in $\mathbb R^n$.
