Definition:Embedded Submanifold

Definition
Let $M$ be a smooth manifold with or without boundary.

Let $S \subseteq M$ be a subset.

Suppose $S$ is a topological manifold in the subspace topology.

Suppose $S$ is endowed with the smooth differentiable structure such that the inclusion map $i_S : S \to M$ is a smooth embedding.

Then $S$ is called the embedded submanifold of $M$.