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 xX, 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.

Read more about this topic:  Covering Space

Famous quotes containing the words formal and/or definition:

    It is in the nature of allegory, as opposed to symbolism, to beg the question of absolute reality. The allegorist avails himself of a formal correspondence between “ideas” and “things,” both of which he assumes as given; he need not inquire whether either sphere is “real” or whether, in the final analysis, reality consists in their interaction.
    Charles, Jr. Feidelson, U.S. educator, critic. Symbolism and American Literature, ch. 1, University of Chicago Press (1953)

    No man, not even a doctor, ever gives any other definition of what a nurse should be than this—”devoted and obedient.” This definition would do just as well for a porter. It might even do for a horse. It would not do for a policeman.
    Florence Nightingale (1820–1910)