# The definition of an interior chart

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 $$\mathbb R^n$$.
• Is $V\subseteq U$ a typo? I think it should be $U\subseteq V$. – Steve 2 days ago