I am reading Lee's Introduction to Topological Manifolds and attempting to prove the following proposition:
Proposition 2.58. If $M$ is an n-dimensional manifold with a boundary, then $\textrm{Int} M$ is an open subset of $M,$ which is itself an n-dimensional manifold without boundary.
Where manifolds with a boundary are defined in terms of charts mapped to open sets in $\mathbb{H}^n = \mathbb{R}^{n-1} \times [0, \infty).$ I need to prove this without using the invariance of the boundary (i.e. that the manifold boundary and interior are disjoint.
My attempt so far involved the construction of charts $(U_i, \varphi_i)$ that cover $M$ and the identification of points mapped to $\partial\mathbb{H}^n$ with $\partial M;$ however, I can't use $\textrm{Int} M = M \setminus \partial M$ without invoking the invariance of boundary. I would prefer hints or partial answers suggesting how I should proceed to full proofs.
It's hard to give a hint to this problem without giving the whole thing away. By definition, $\text{Int} M$ is the subset of all $x \in M$ for which there exists a chart $(U_i,\phi_i)$ such that $x \in U_i$ and such that $\phi_i(U_i) \subset \mathbb R^{n-1} \times (0,\infty)$. This existence property is clearly also true for every $y \in U_i$, using the exact same chart $(U_i,\phi_i)$, and therefore $U_i \subset \text{Int}(M)$.