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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!

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  • $\begingroup$ 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. $\endgroup$ – Mariano Suárez-Álvarez May 3 '16 at 2:34
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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)

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