I'm trying to prove the existence of Morse functions on differentiable manifolds, by adapting the proof found on Matsumoto's textsbook, which works for compact manifolds, to the non-compact case.
I think I found a way to use paracompactness instead of compactness to prove the assertion for general manifolds, but I still need the existence of particular covers for the proof to work.
Let $M$ be a differentiable manifold, I need to prove the existence of two covers $\mathcal{U}$ and $\mathcal{K}$ such that
$\mathcal{U}$ and $\mathcal{K}$ are at most countable, write $\mathcal{U} = \{U_1, U_2, \ldots, U_i, \ldots \}$ and $\mathcal{K} = \{K_1, K_2, \ldots, K_i, \ldots \}$ and $K_i \subseteq U_i$ for all $i$
$\mathcal{U}$ is made up of open sets homeomorphic to open euclidean sets
$\mathcal{U}$ is locally finite
Sets in $\mathcal{K}$ are compact
It is easy to get all conditions working for $\mathcal{U}$, and it is also straightforward to obtain a countable compact cover $\mathcal{K}$ which is a refinement of $\mathcal{U}$, but I can't get $\mathcal{K}$ to have the property that only a finite number of $K_i$'s are contained in each $U_j$ (if this were to hold we would be done, by considering the union of such $K_i$'s, defining a new compact cover formed by such unions, getting rid of redundant $U_j$'s and reordering the indices).
Any help in proving the assertion (or finding a counterexample to it) is greatly appreciated. Thank you :)