Let $M$ differential manifold. Show that exist a cover set $\{U_{\alpha}\}$ of $M$ with the following propierties: $$U_{a}\mbox{ is a open "contractible", for each } \alpha $$ $$\mbox{If } U_{\alpha_{1}},...,U_{\alpha_{n}} \mbox{ are elements of the cover set, then } \bigcap_{i=1}^{r}{U_{\alpha_{i}}}\mbox{ is contractible}$$
A subset $A$ of the differential manifold $M$ is contractible to the point $a\in A$ when, the aplications $id_{A}$ (identity in $A$) and $\kappa_{a}:x\in A\to a\in A$ are homotopic (with a point basis $a$). Any hint, thanks!
I weakly remembered that in 1962 or so I had read a paper by André Weil proving exactly this. And indeed, here it is the reference:
https://en.wikipedia.org/wiki/Good_cover_(algebraic_topology)