If $\phi : \partial{M} \times [0,1) \to M $ is an embedding prove that $\phi(\partial{M} \times [0,a])$ is closed in $M$.

Question

Let $M$ be a compact manifold and let $\phi:\partial{M}\times [0,1) \to M $ be an embedding whose image is open in $M$. How to prove that $ \phi(\partial{M} \times [0,a]) $ is closed in $M$ for any $a>0$.

I was studying adjunction spaces, and, I want to prove the hausdorffness of the adjunction space. I need to prove this fact first in order to get the open sets separating any two distinct points in that space. How to prove this fact. Please help..