Vector bundle

A vector bundle is a space assembled from vector spaces attached continuously to the points of a base space. Locally, it resembles a product of the base with a fixed vector space.

Fiber bundle: A space locally modeled on a product of a base space and a typical fiber. A vector bundle is a fiber bundle whose fibers carry compatible vector-space structures.
Frame bundle: The bundle whose fiber at each point consists of ordered bases of the tangent space there. Its local frames encode coordinate changes that govern associated vector bundles.
Tangent bundle: The bundle whose fiber at each point of a manifold is its tangent space. It packages all tangent directions into one vector bundle over the manifold.
Trivial vector bundle: A vector bundle isomorphic to a product of its base with a fixed vector space. It has no global twisting, unlike vector bundles that are locally but not globally products.
Topological space: A set equipped with a topology that specifies which subsets are open. The base and total spaces use topology to express continuity and local resemblance.
Clutching function: A map on an overlap or boundary used to glue pieces of a bundle together. Clutching functions construct bundles by prescribing how local pieces are joined.
Cotangent bundle: The bundle whose fiber at each point is the dual of the tangent space. It provides the natural home for differential one-forms and covectors.
Möbius strip: A nontrivial real line bundle over the circle, formed by a half-twist. It is the simplest familiar example of a vector bundle that is not a product.
Fiber: The preimage of a point under a map, often treated as the space lying over that point. Each point of the base has a vector space as its fiber.
Associated bundle: A bundle formed from a principal bundle and a group action on a typical fiber. It builds vector bundles from symmetry data and representations.

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