A vector bundle is a mathematical structure that consists of a collection of vector spaces attached to each point of a topological space. This means that for every point in the space, there is a corresponding vector space, allowing for the study of functions and fields that vary smoothly across the space. Vector bundles are essential in various areas of mathematics, including geometry and topology. One common example of a vector bundle is the tangent bundle, which associates a vector space of tangent vectors to each point on a manifold. This concept is crucial in the study of differential geometry and is used in physics, particularly in the theory of general relativity and gauge theories.