Manifold Neighbourhoods and a Conjecture of Adjamagbo

We verify a conjecture of P. Adjamagbo that if the frontier of a relatively compact subset $V_0$ of a manifold is a submanifold then there is an increasing family $\{V_r\}$ of relatively compact open sets indexed by the positive reals so that the frontier of each is a submanifold, their union is the whole manifold and for each $r\ge 0$ the subfamily indexed by $(r,\infty)$ is a neighbourhood basis of the closure of the $r^{\rm th}$ set. We use smooth collars in the differential category, regular neighbourhoods in the piecewise linear category and handlebodies in the topological category.