A topological space C together with a continuous surjective map p: C → X (the covering map) onto a base space X such that every point of X has an open neighborhood U for which p^{-1}(U) is a disjoint union of open sets in C, each of which is mapped homeomorphically onto U by p. The fibers p^{-1}(x) are discrete sets and p is a local homeomorphism.

Principle

A covering space organizes a global projection by locally trivializing the projection into a disjoint union of identical local sheets; the covering map is a local homeomorphism with discrete fibers and so encodes how local copies of the base are assembled into a possibly more highly connected total space.

Demonstration

The exponential map R → S^1, t ↦ e^{2πit}, is a covering map: each small arc U ⊂ S^1 is evenly covered and its preimage is a disjoint union of intervals in R. More generally, the n-fold map S^1 → S^1, z ↦ z^n, yields an n-sheeted covering; the universal cover of a wedge of circles can be drawn as an infinite tree that projects to the wedge.

Misapplication

Treating any surjective local homeomorphism or any map with discrete fibers as a covering without checking the evenly covered neighborhood condition, or calling branched or ramified maps (which have points with non‑evenly covered neighborhoods) coverings. Also conflating covering spaces with arbitrary fiber bundles when the fiber is not discrete.

Consequence

Coverings have the unique path-lifting and homotopy-lifting properties: paths and homotopies in the base lift to the cover once a starting point in the appropriate fiber is chosen. Classification of coverings of a connected, locally path-connected, semilocally simply connected space corresponds to conjugacy classes of subgroups of its fundamental group.

Reversal

The inverse notion is taking a quotient that identifies distinct points in fibers to produce the base (for example, quotienting a universal cover by the action of a deck group). Reversal highlights passing from a highly connected total space down to a space with more identification and possibly nontrivial fundamental group.

Boundary

Definition requires a continuous surjective map and the evenly covered neighborhood condition; it excludes branched covers, ramified maps, maps between spaces lacking local path-connectedness or the discreteness of fibers, and general fiber bundles whose fibers are not discrete sets. The base is typically assumed to be path-connected when discussing classification.

Semantic Tension

Confusion often arises between 'covering space' and 'fiber bundle': both locally look trivial, but coverings have discrete fibers and the evenly covered condition. Another nearby notion is 'local homeomorphism', which is necessary but not alone sufficient in pathological bases without evenly covered neighborhoods.

Synthesis

A covering space is a total space mapped onto a base by a local homeomorphism that globally arranges discrete, identical local sheets; it is the context in which path- and homotopy-lifting work and in which algebraic invariants of the base (subgroups of π1) classify possible covers.