Coveringsuch

Coveringsuch is a term used in topology to describe an augmentation of the standard covering space concept with an additional selection rule, intended to model how particular choices are carried along the base space.

Formally, let p: E → B be a covering space. A covering such, denoted σ, assigns to every point b ∈ B a nonempty finite subset σ(b) ⊆ p^{-1}(b). The assignment is locally constant: for each b there is a neighborhood U with a local trivialization p|_{p^{-1}(U)} ≅ U × F such that for every b' ∈ U, σ(b') is carried to a fixed finite subset of F by the local identifications. Moreover, σ is compatible with monodromy: following a loop based at b permutes the elements of σ(b) in a way that depends only on the loop’s homotopy class. If a global σ can be chosen for all b, the covering is said to admit such a selection.

Examples and remarks: In a trivial covering where E ≅ B × F, one can take σ(b) = {b} × S for a fixed finite subset S ⊆ F; this provides a simple instance of a covering that carries a global σ. For nontrivial coverings with nontrivial monodromy, a global σ often does not exist, illustrating how global choices interact with topological structure.

Applications and relation: Coveringsuch serves as a theoretical tool to discuss lifting problems, monodromy actions, and the relationship between local triviality and global constraints. It sits alongside traditional covering space theory and related ideas in sheaf theory and local-to-global methods.

See also: Covering space, Monodromy, Sheaf theory, Section.