Immersion of the manifold in the Flowout Theorem I am reading John Lee "Introduction to Smooth Manifolds." Flowout theorem states (Theorem $9.20$, Flowout Theorem): 
Suppose $M$ is a smooth manifold, $S\subset M$ is and embedded $k$-dimensional sumbanifold,and $V$ is a smooth vector field that is nowhere tangent to $S$. Let $\theta:D\rightarrow M$ be the flow of V, let $O = (\mathbb{R} \times S) \cap D$ and let $\Phi = \theta|_O$ then there exist positive function $\delta: S \rightarrow \mathbb{R}$ such that the restriction of $\Phi$ to $\{(t,p)\in O:|t|<\delta(p)\}$ is injective. In particular $\Phi(\{(t,p)\in O:|t|<\delta(p)\})$ is an immersed sumbanifold. 
My question is: why does it not an embedded one? It seems to be an embedded manifold for the case when $S$ has codimension $1$. I was trying to come up with an example but failed. 
 A: One can use Example 4.20 to show that there exist flowouts (where more generally the domain is an open subset of $\mathbb R \times S$ containing $\{0\} \times S$), which are not embedded:
As in the book, let 
$$M:= \mathbb{T}^2 = \left\{ \left( z_1, z_2 \right) \in \mathbb {C}^2
\middle\vert \, \left\lvert z_1 \right\rvert  = \left\lvert z_2 \right\rvert  =1  \right\}$$ be the torus and $\alpha$ be an irrational number. By identifying the tangent bundle of $\mathbb{C}^2$ with $\mathbb C^4$ in the obvious manner, we may consider the vector field 
$$\mathbb{C}^2 \to \mathbb{C^4} \quad\colon \quad \left( z_1, z_2\right)
\to \left( z_1, z_2, 2 \pi i \, z_1, 2 \pi i \alpha \, z_2 \right) \, ,$$
which restricts to a smooth vector field $V$ on $M$. The set $S:= \{ (1,1)\}$ is a $0$-dimensional embedded submanifold of $M$. Any non-zero vector at the point is nowhere tangent to $S$ and so is $V$. Now check that 
$$\gamma \colon \quad \mathbb R \to M \quad\colon \quad t \to 
\left( e^{2 \pi i \, t},  e^{2 \pi i \alpha \, t}\right)$$
is an integral curve of $V$ and defines a flowout from $S$ along $V$ in the sense that $\gamma$, considered as a map defined on $\mathbb R \times S$, is an immersed submanifold. However, $\gamma$ is not a topological embedding, as its image is dense in $M$. 
Yet if you restrict its domain to say $(-1,1) \times S$, then it is indeed an embedded submanifold. 
My intuition tells me that, even in the general case, making the neighborhood around S 'small enough' (i.e. choosing a different function $\delta'$ on $S$ with $0< \delta' \leq \delta$) should yield embeddedness, but I have not been able to turn this into a rigorous argument yet. The idea behind this is that embeddedness tends to fail either because of similar issues as in the example above or when the manifold `touches itself in the wrong way' (see figure 4.3 in the book). 
