To each point P of a smooth manifold M is associated a tangent space TP(M). The union of these tangent spaces
can itself be given the structure of a manifold. TM is known as the tangent bundle and has twice the dimension of M. There is a natural (first coordinate) projection π:TM → M and TP(M) = π–1(P ) is called the fibre over P.
Tangent bundles are examples of vector bundles which more generally are manifolds comprising vector spaces (the fibres) associated with points of the base manifold M. The topology of a manifold can be investigated by the use of vector bundles and other spaces naturally associated with the manifold.