In mathematics, an open cover of a topological space
X
is a family 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 in the cover and all intersections of finitely many open sets,
U
α
1
…
α
n
=
U
α
1
…
α
n
−
1
∩
U
α
n
, are contractible (Petersen 2006).
The concept was introduced by Andre Weil in 1952 for differential manifolds, demanding the
U
α
1
…
α
n
to be differentiably contractible. A modern version of this definition appears in Bott & Tu (1982).
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.)
The two-dimensional surface of a sphere
S
2
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.
