Abstract: PROOF. The long line L is defined in Hocking and Young [1], for example. It is easy to check that L, with the topology defined there, supports the structure of a CX manifold. In fact, it is proved in [2] that L (there called the Alexandroff Half Line) can be made into a real analytic manifold (I thank the referee for this reference). Let us recall what it means for a real vector bundle E to be trivial. The bundle E2i-X is trivial if there is a bundle map j: E-*+n covering the identity: X-*X, where (n is the trivial Rn bundle over X and n is the dimension of the fibre of E. But this is easily seen to be equivalent to the existence of a bundle map f: E->Rn where Rn is considered as a bundle over a point *. (If w: {n-4Rn is projection onto the coordinate of each fibre, then f =w o q4 is such a map. If + is given then f-'Rn = in and Ei--hRn where f-' means the pull back under f.) So let TL be the tangent bundle of L and 0 be a Cx trivialization