Show that exist a “good” cover $\{U_{\alpha}\}$ of the Differential Manifold

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!

• If you search in MathOverflow you'll find one of my questions which had a fee answers explaining a couple of approaches to do this. – Mariano Suárez-Álvarez May 3 '16 at 2:34