(a) Let A C Rn be an open set such that boundary A is an (n - 1) -dimensional manifold. Show that N = AU boundary A is an -dimensional manifold with boundary. (It is well to bear in mind the following example: if A = {x ЄRn}: |x| < 1 or 1 < |x| < 2}, then N = AU boundary A is a manifold with boundary, but ∂ N ≠ boundary A.
(b) Prove a similar assertion for an open subset of an n-dimensional manifold.