Good cover (algebraic topology)

In mathematics, an open cover of a topological space ${\displaystyle X}$ is a family of open subsets such that ${\displaystyle X}$ 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 (Petersen 2006).

The concept was introduced by André Weil in 1952 for differentiable manifolds, demanding the ${\displaystyle U_{\alpha _{1}\ldots \alpha _{n}}}$ to be differentiably contractible. A modern version of this definition appears in Bott & Tu (1982).