Let $M$ be an n-dim topological manifold with boundary, A chart for M is a pair $(U,\phi)$ where $\phi: U\rightarrow \mathbb{R}^n$ is a map such that $\phi$ is a homeomorphism onto an open subset of $\mathbb{H}^n$ or $\mathbb{R}^n$. If $p\in M$ is not a boundary point, in which case, restricting the coordinate map to $\phi^{-1}{(int\mathbb{H}^n)}$ is an interior chart whose domain contains p.
My question is: Isn't the author assuming that $\phi$ is continuous? Notice, he sais: "$\phi: U\rightarrow \mathbb{R}^n$ is a map such that $\phi$ is a homeomorphism onto an open subset of $\mathbb{H}^n$ or $\mathbb{R}^n$." I know that homeomorphisms are continuous.If $\phi$ is continuous... then can I say that $\phi^{-1}(int\mathbb{H}^n)$ is open in $U$.