Let $M$ be a topological manifold which is homeomorphic to the interior of a compact manifold with boundary. This extends the class of compact manifolds and is sometimes called being of finite topological type.
What can be said about the CW-complex of $M$?
For example, I'm concerned about the following question:
Does $M$ have the homotopy type of a CW-complex of finite type, i.e. a CW-complex whose skeletons are finite?
If I remember correctly, topological manifolds have the homotopy type of CW-complexes and compact manifolds have the homotopy type of finite CW-complexes. So this would fit the picture.
This was discussed several times at MSE (at least here and here). The simplest (still quite hard) proof that each compact manifold is homotopy equivalent to a finite CW complex is due to Chapman, see my answer here. By Brown's theorem, every topological manifold $M$ with boundary is collarable (the boundary admits a product neighborhood, see my answer here) it follows that $int(M)$ is homotopy-equivalent to $M$. See also Grumpy's answer here in the smooth case.