I want to understand if it is always possible to find the following on a simply connected manifold $M$:
A countable family $\{U_{n}\}_{n=1}^{\infty}$ of simply connected open subsets with compact closure such that
- $U_{n}\subset U_{n+1}$ $\forall n\in\mathbb{N}$,
- $\bigcup_{n=1}^{\infty}U_{n} = M$.
Maybe this can be done by finding an adequate Moser function $f:M\rightarrow\mathbb{R}$.
In particular, I am interested in doing this for any symplectic manifold $(M,\omega)$.
Examples of manifolds admitting such exhaustions are $\mathbb{R}^{n}$ and compact simply connected manifolds (which no need to be symplectic).
Do you have any counterexample? Or, do you know a larger family of (symplectic) manifolds for which this construction holds?
Thank you.
Pd: Just to be precise, my definition of simply connected implies connected.