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 of a manifold, which consists of all the tangent spaces at each point of the manifold. This structure is crucial in differential geometry and is used to analyze the properties of curves and surfaces. Vector bundles also play a significant role in physics, particularly in the formulation of gauge theories and the study of fields.