Separating submanifold

Definition
A submanifold of a connected manifold is termed separating if it has codimension 1, and if its complement in the manifold is disconnected.

Facts

 * In a simply connected manifold, any compact connected submanifold of codimension one, is separating.

(In fact much weaker assumptions that simple connectedness suffice; all we need is that the first homology has no 2-torsion).