There are three types of submanifolds discussed in my book. Let $M$ be a smooth manifold. Then
1.) An immersed submanifold of $M$ is a set $S\subseteq M$ such that $S=F(S)$, where $F:N\to M$ is an injective immersion, with $S$ given the final topology and the smooth structure: $(U,\phi)$ is chart in $N\Leftrightarrow$ $(F(U),\phi\circ F^{-1})$ is a chart on $S$.
2.) An embedded submanifold of $M$ is an immersed submanifold where the final topology of $S$ is the subspace topology of $S$ in $M$. An embedding is an injective immersion $F:N\to M$ such that $F(N)$ is homeomorphic to $N$, where $F(N)$ has subspace topology. Thus, embedded submanifolds of $M$ are precisely the images of embedding $F:N\to M$
3.) A regular submanifold of dimension $k$ in $M$ is a subset $S\subseteq M$ such that $\forall s\in S$ there exists a chart $(U,\phi)$ of $M$ containing $s$ such that $U\cap S=\phi^{-1}(\{(x_1,...,x_n)\in\mathbb{R}^n:x_{k+1}=...=x_n=0\})$. It can be shown, that if $F:N\to M$ is an embedding, then $F(N)$ is a regular submanifold of $M$. Thus, every embedded sumbanifold is a regular submanifold.
Question: Let $I_M$, $E_M$, and $R_M$ be the sets of immersed submanifolds of $M$, embedded submanifolds of $M$, and regular submanifolds of $M$, respectively. We have $E_M\subset I_M$ and $E_M\subset R_M$, but is there a relationship between $R_M$ and $I_M$?
