is a vector space isomorphism of Vn onto T(m). The tangent bundle 3(M) of the manifold M consists of the ordered pairs (m, v) where meM and vE T(m). Therefore, as a point set only, 3(M) is M x V1. We shall assume that the reader is familiar with the fibre space topology which is customarily assigned to 3(M). (For a description of this topology and the facts concerning fibre bundles which we shall need in other parts of this paper see N. Steenrod, The topology of fibre bundles, Princeton, 1951.) For many manifolds M this topology differs from the topology of the product space M X Vn. This fact indicates that it is the assignment of this topology which makes 3(M) an interesting mathematical object. It is the purpose of this paper to present an intrinsic characterization of this topology.