$M\setminus\partial M$ open and proof of dimension

Question

Let $M$ be an $n$-dimensional manifold with boundary. Show that:

1. $M\setminus\partial M$ is an open subset of $M$ and it is an $n$-dimensional manifold, and that

2. $\partial M$ is a closed subset of $M$ and it is an $n-1$-dimensional manifold.

I know that $\partial M$ being closed follows from $M\setminus\partial M$ being open, but I don't know how to show it's open. I also have no idea how to prove the dimension of a manifold because I've never worked with manifolds before.

Answer 1

Some hints.

For 1., notice that an open subset is always a submanifold of the same dimension. To see that $M\setminus\partial M$ is open, just note that the trace in any map is open.

For 2., look at the maps of $M$ near the boundary and just restrict them.