Connected manifold with disconnected boundary?

Question

Is there any simple example of a connected manifold with disconnected boundary?

Answer 1

Not only are there many connected manifolds with disconnected boundaries (such as $[0,1]$ or a cylinder), there is an equivalence relation built on it called cobordism, where two manifolds are said to be cobordant if their disjoint union is the boundary of a manifold one dimension higher. Technically, the higher dimensional manifold need not be connected, but this is often the case.