Covering Space - Formal Definition

Formal Definition

Let X be a topological space. A covering space of X is a space C together with a continuous surjective map

such that for every x ∈ X, there exists an open neighborhood U of x, such that p−1(U) (the inverse image of U under p) is a disjoint union of open sets in C, each of which is mapped homeomorphically onto U by p.

The map p is called the covering map, the space X is often called the base space of the covering, and the space C is called the total space of the covering. For any point x in the base the inverse image of x in C is necessarily a discrete space called the fiber over x.

The special open neighborhoods U of x given in the definition are called evenly-covered neighborhoods. The evenly-covered neighborhoods form an open cover of the space X. The homeomorphic copies in C of an evenly-covered neighborhood U are called the sheets over U. One generally pictures C as "hovering above" X, with p mapping "downwards", the sheets over U being horizontally stacked above each other and above U, and the fiber over x consisting of those points of C that lie "vertically above" x. In particular, covering maps are locally trivial. This means that locally, each covering map is 'isomorphic' to a projection in the sense that there is a homeomorphism from the pre-image of an evenly covered neighbourhood U, to U X F, where F is the fiber, satisfying the local trivialization condition. That is, if we project this homeomorphism onto U (and thus the composition of the projection with this homeomorphism will be a map from the pre-image of U to U), the derived composition will equal p.