Let $M$ be an oriented manifold (with boundary) with $dim (M)\ge 2$. Show that there exists an atlas $\{(U_{\alpha},\phi_\alpha)\}_{\alpha\in I}$ for the chosen orientation such that $\forall\alpha\in I$ we have $\phi_\alpha(U_\alpha)$ is open in $\mathbb{R}^n$ or in $\mathbb{R}^n_+$.
Let's choose an orientation, i.e. let's choose a maximal oriented atlas $\{(U_{\alpha},\phi_\alpha)\}_{\alpha\in I}$.
I know that $$ \mathring{M}:=\{p\in M:\exists (U,\phi), p\in U, \phi(U)\,\,\text{open}\,\, \text{in}\,\, \mathbb{R}^n \}\\ \partial M=\{p\in M:\forall (U,\phi),p\in U,\phi(p)\in\partial\mathbb{R}^n_{\pm}\} $$ I don't know how to proceed