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Let $M$ be a simply-connected smooth manifold with dimenison $n$.

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

math student
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  • Google "good cover". – Moishe Kohan Feb 15 '20 at 04:32
  • Isn't it obvious, from the definition of manifold? – Dong-gyu Kim Feb 15 '20 at 05:07
  • Note that $\R^n$ is contractible, hence every locally euclidean space has a cover by contractible neighborhoods. The good covers mentioned by Moishe Kohan have the further property that intersections of the covers are either empty or contractible as well. Their existence is more dubious to me than that of the cover you originally asked for... – Jonas Linssen Feb 15 '20 at 05:08
  • So, in other word, any open covering is contractible? – math student Feb 15 '20 at 06:14
  • No. open covers need not be contractible. That is in no way a consequence of what PrudiiArca said. For example, the circle is an open cover of itself, and is not contractable. – Paul Sinclair Feb 15 '20 at 17:51

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