Separating submanifold

This article defines a property of a submanifold inside a manifold

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. For full proof, refer: compact connected implies separating in simply connected

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

• In a simply connected manifold, any compact connected separating submanifold is orientable