Let $M$ be a smooth connected manifold with boundary. Is it true that the interior of $M$ is also connected? (As usual I am assuming $M$ is second-countable and Hausdorff).
I am trying to rule out something like two "tangent disks" (or circles) where the (topological) interior is obviously not connected. This case is not a counter-example since the union of two tangent disks (or circles) is not a manifold with boundary (the point of tangency is pathological).