I try to construct an example of a partition of unity on a manifold M, s.t. at least one function $\displaystyle \psi_j \neq 0$ and supp $\displaystyle \psi_j$ is not compact.

First I try to construct two functions, s.t. their sum is equal 1. But i couldn't find a example, s.t. the support is not compact. We know it is closed.

But on the other hand there are manifolds, which are compact. And every closed subset of a compact space is compact itself. So there can't be a general example for all manifolds. Therefore, i have to give some special example with a special manifold.
Are these ideas correct?

Then i can just take the real line with the discrete topology and put one function which is identically 1 everywhere. Since the real line isn't compact, its done!

But I'm not sure, whether there really isn't some general example (partition of unity) for any manifold...