Finitesheeted

Finitesheeted, or finite-sheeted, is a term used in covering space theory to describe a covering map p: E -> B whose fibers are finite sets. Concretely, for every b in B, the fiber p^{-1}(b) is finite. If the base space B is connected, the cardinality of the fibers is constant, and this common value is called the degree or the number of sheets of the covering, commonly denoted n. Thus p is an n-sheeted covering.

In the standard setting of algebraic topology—where B is path-connected, locally path-connected, and semilocally simply connected—finite-sheeted covers correspond to subgroups of finite index in the fundamental group π1(B). The degree n equals the index [π1(B): p_*(π1(E))]. A cover is connected exactly when the associated subgroup has finite index and acts transitively on the fiber; in general, a finite-sheeted cover can have several connected components, each a connected covering of B, with the total degree equal to the sum of the degrees of its components.

Key properties include closure under composition: the composition of finite-sheeted coverings is finite-sheeted. The pullback of a finite-sheeted cover along any map is finite-sheeted. If B is connected, the total space E has finitely many connected components, and each component is a connected finite-sheeted cover of B.

Examples include the n-fold covers of the circle S^1, such as p: S^1 -> S^1 with p(z) = z^n, which is an n-sheeted cover. By contrast, the universal cover R -> S^1 is infinite-sheeted. Finite-sheeted covers play a central role in relating topological coverings to subgroup structure of fundamental groups.