{{Short description|Open topological cover}} thumb|The cover on the left is not a good cover, since while all open sets in the cover are contractible, their intersection is disconnected. The cover on the right is a good cover, since the intersection of the two sets is contractible. In mathematics, an open cover of a topological space <math>X</math> is a family of open subsets such that <math>X</math> is the union of all of the open sets. A '''good cover''' is an open cover in which all sets and all non-empty intersections of finitely-many sets are contractible {{harv|Petersen|2006}}.
The concept was introduced by André Weil in 1952 for differentiable manifolds, demanding the <math>U_{\alpha_1 \ldots \alpha_n}</math> to be differentiably contractible. <!-- diffeomorphic to the <math>n</math>-dimensional Euclidean space <math>\mathbb{R}^n</math>.--> A modern version of this definition appears in {{harvtxt|Bott|Tu|1982}}.
==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 with a good cover is the same as the Čech cohomology of the space. (Such a cover is known as a Leray cover.) However, for the purposes of computing the Čech cohomology it suffices to have a more relaxed definition of a good cover in which all intersections of finitely many open sets have contractible connected components. This follows from the fact that higher derived functors can be computed using acyclic resolutions.
==Example==
The two-dimensional surface of a sphere <math> S^2 </math> has an open cover by two contractible sets, open neighborhoods of opposite hemispheres. However these two sets have an intersection that forms a non-contractible equatorial band. To form a good cover for this surface, one needs at least four open sets. A good cover can be formed by projecting the faces of a tetrahedron onto a sphere in which it is inscribed, and taking an open neighborhood of each face. The more relaxed definition of a good cover allows us to do this using only three open sets. A cover can be formed by choosing two diametrically opposite points on the sphere, drawing three non-intersecting segments lying on the sphere connecting them and taking open neighborhoods of the resulting faces.
==References== *{{citation | last1 = Bott | first1 = Raoul | authorlink = Raoul Bott | last2=Tu |first2= Loring | title = Differential Forms in Algebraic Topology | year = 1982 | publisher = Springer | location = New York | isbn = 0-387-90613-4}}, §5, S. 42. *{{citation | last = Weil | first = Andre | authorlink = Andre Weil | title = Sur les theoremes de de Rham | year = 1952 | journal = Commentarii Math. Helv. | volume = 26 |pages=119–145| doi = 10.1007/BF02564296 | s2cid = 124799328 }} *{{citation | last = Petersen | first = Peter | edition = 2nd | isbn = 978-0387-29246-5 | mr = 2243772 | page = 383 | publisher = Springer | location = New York | series = Graduate Texts in Mathematics | title = Riemannian geometry | url = https://books.google.com/books?id=9cekXdo52hEC&pg=PA383 | volume = 171 | year = 2006}}
Category:Algebraic topology Category:Homology theory