Covering of differentiable manifolds with open disks

Summary The aim of this paper is to prove that every open (i.e. noncompaet without boundary) manifold of dimensionn can be covered with exactlyn open disks. This is a generalization of a theorem of E. Luft 3] concerning the case of any open 2-dimensional manifold. It is then proved that every compact manifold of dimensionn with nonempty boundary can also be covered with exactlyn open disks. The proofs of the theorems are in the spirit of Morse theory 1].