Coxeter notation

In mathematics, an open cover of a topological space $X$ is a set of open subsets such that $X$ is the union of all of the open sets. In algebraic topology, an open cover is called a good cover if all open sets and all multiple intersections, $U_{\alpha _{1}\ldots \alpha _{n}}=U_{\alpha _{1}\ldots \alpha _{n-1}}\cap U_{\alpha _{n}}$, are contractible.Potter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park.

The concept was introduced by Bott and Tu on differential manifolds, demanding the $U_{\alpha _{1}\ldots \alpha _{n}}$ to be diffeomorphic to the $n$-dimensional Euclidean space $\mathbb {R} ^{n}$.

Application

A major reason for the notion of a good cover is that the Leray spectral sequence of a fiber bundle degenerates for a good cover, and so the Čech cohomology associated to a good cover is the same as the Čech cohomology of the space. (Such a cover is known as a Leray cover.)

Example

To cover the two sphere $S^{2}$ with a good cover one needs at least 4 open sets.

References

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