Any manifolds have an open covering such that each is constractible?

Question

Let M be an arbitrary smooth manifold.

Then, is there an open covering $\{U_i\}$ of $M$ such that each $U_i$ is constractible anytime?

Answer 1

Yes! This comes almost from the definition.

When we define the manifold structure on $M$, we choose an open cover $\{U_i\}$ and diffeomorphisms from the $U_i$ to an open set in $\mathbb{R}^n$ (with extra conditions).

One thing we could do is require that the $U_i$ map diffeomorphically to the open unit ball. This means that each $U_i$ is contractible, and they form a cover by assumption.