vezelbundels

Vezelbundels, or fiber bundles, are a mathematical framework for spaces that locally look like a product but may have a twisted global structure. A vezelbundel consists of a total space E, a base space B, a typical fiber F, and a continuous surjection p: E → B. For each b in B there is a fiber F_b = p^{-1}(b). The data is required to be locally trivial: there exists an open cover {U_i} of B and homeomorphisms φ_i: p^{-1}(U_i) → U_i × F such that the projection to U_i matches p. On overlaps U_i ∩ U_j the transition functions g_{ij}: U_i ∩ U_j → Homeo(F) describe how the local product pieces are glued together and satisfy a cocycle condition. The structure group G is a subgroup of Homeo(F) generated by these transition functions, and the bundle is often described as a G-bundle when the changes of coordinates lie in G.

If the fiber F is a vector space and all transition maps are linear, the result is a vector bundle; if the fiber carries a group action, one speaks of a principal bundle. Prototypical examples include the tangent bundle TM of a differentiable manifold M, whose fibers are tangent spaces; the Möbius strip, a nontrivial line bundle over the circle S^1; and the trivial bundle B × F, which is globally a product. Fiber bundles provide a unifying language for geometric and topological constructions where local product structure is preserved while global twisting varies.