One of my professors assigned this problem last semester and I recently looked back at it and am still a bit confused. Let $M$ be a topological n manifold with boundary, which means, a second-countable, Hausdorff space such that for every $x \in M$, there is a neighborhood $U \ni x$ which is homeomorphic to either $\mathbb{R}^{n}$ or $\mathbb{R}^{n-1} \times [0,\infty)$. Also, assume that $\partial M$ has countable many components.
We have to prove that the inclusion $i : M - \partial M \to M$ is a homotopy equivalence.
I guess if one assumes compactness, you can use the Proposition 3.42 of Hatcher to get a collar neighborhood of $\delta M$ but I am not sure how to proceed from there, if this approach even works.
Once you have a collar neighborhood $N(\partial M) \approx \partial M \times [0,1)$ in your hands, you have to guess how to use it to define a homotopy inverse $g : M \to M - \partial M$.
But that's pretty natural. First let $k : [0,1) \to [.5,1)$ be a homeomorphism which is constant on $[.75,1)$. Outside of the collar neighborhood define $g$ to be the identity. Inside the collar neighborhood, define $g$ to be the composition the map $$\partial M \times [0,1) \xrightarrow{\text{Id} \times k} \partial M \times [.5,1) \hookrightarrow \partial M \times (0,1) $$
And finally you have to prove that the map $g : M \to M-\partial M$ is homotopic to the identity. But again that's done in pretty much the same way, starting with a homotopy from the composition $$[0,1) \xrightarrow{k} [.5,1) \hookrightarrow [0,1) $$ to the identity in such a way that the homotopy is stationary on $[.75,1)$.