Let $M$ be compact manifold of dimension $n$. I am wondering if a certain type of covering of M by coordinate charts exists (this is not the usual good covering theorem but seems related to it). I want to have a cover of M by finitely many closure of open coordinate balls $\{\bar{U}_i\}$ s.t
i. they only intersect at the boundary
ii. the intersections consists of union of finite number of embedded submanifolds of lower dimension.
(it would be even nicer if in ii. these submanifolds dont accumulate on each other. this does not seem essential but this requirement would for instance rule out triangulations)
The well known "good covering" theorem seems to be close to this in the sense that if you contract the intersections then the open covers only intersect at the boundaries. You can break each open ball into coordinates balls so that is not a problem but I am still not sure if the intersections of these covers are embedded manifolds.
I also know that if you have a measure $\mu$ on M, you can choose such a covering by positive measure sets that they intersect only at measure zero sets. But this is again not enough as measure zero set could be anything.
For instance you can do this definitely in $S^n$ in $T^1,T^2,T^3$ and probably in $T^n$ although I havent thought about it.
Any reference answering this question is also very welcome.
Hint: Use the fact that every smooth manifold admits a triangulation, see here for a simple proof.