Introduction. LetXbe a submanifold ofY, in either the topological, smooth, or piecewise linear ( =PL) categories. Anormal cell bundleonXinYis a bundle ξ = (p, E, X) in the category whose fibre is a closed cell, and such thatEis a neighbourhood ofXinYandp: E → Xis a retraction. The triple (Y, X, ξ) is atubular neighbourhood, or briefly, atube. For convenience we may refer to a tube by its cell bundle.