Embedded submanifold

An embedded submanifold $M\subseteq N$ is a subset whose inclusion is an embedding. Around every point there are coordinates on $N$ in which $M$ is a coordinate plane.

If embedded submanifolds $A,B\subseteq M$ meet transversely, then $A\cap B$ is an embedded submanifold of dimension $\dim A+\dim B-\dim M$.