If $\mathcal{O}=(U_{\alpha})_{\alpha\in A}$ is an open cover for a smooth manifold $M$, then each $U_{\alpha}$ is a smooth manifold.
I want to extend this fact to manifolds without boundary. So my question is this (I feel like it is right but would like someone with more authority than me to confirm this):
If $\mathcal{O}=(U_{\alpha})_{\alpha\in A}$ is an open cover for a smooth manifold with boundary $M$, then can $\mathcal{O}$ be decomposed in to two sets $(V_{\beta})_{\beta\in A}$ and $(W_{\gamma})_{\gamma\in A}$ such that each $V_{\beta}$ is a smooth manifold and each $W_{\gamma}$ is a smooth manifold with boundary and all of the sets $V_{\beta}$ and $W_{\gamma}$ are elements themselves of $\mathcal{O}$?
Hopefully my question makes sense, and if it doesn't, someone bright can actually pick up on what I am meant to be asking. Thanks in advance!
EDIT: I think the statement is true. Just decompose $\mathcal{O}$ in to sets containing boundary points and sets containing only interior points. Is there a precise, rigorous way to write this decomposition?