Show that Hopf foliation is a foliation. 
Consider $S^3 := \{(z,w) \in \mathbb{C}^2:|z|^2 + |w|^2 = 1\}$ be the
  unit $3$-sphere with equivalence relation
$$(z,w) \sim (z',w') \iff z' = e^{i \theta }z, w' = e^{i\theta} w$$
  for some $\theta \in \mathbb{R}$.

My definition of foliation:

A rank $k$-foliation of a manifold $M$ is a collection
  $\{L_\alpha\}_{\alpha\in A}$ of connected immersed submanifolds of $M$
  such that
(1) $M= \coprod_{\alpha \in A} L_\alpha$
(2) For every point $p \in M$, there is a chart $(U, \phi=(x^1, \dots,
 x^m))$ with $p \in U$ such that for every leaf $L_\alpha$ we have that
  $U \cap L_\alpha$ is empty or the countable union of slices of the
  form $\{x^{k+1}= constant, \dots, x^m = constant\}$.

Now, I see that 
$$[(z,w)] = \{e^{i \theta}(z,w): \theta \in \mathbb{R}\}$$
but I cannot proceed after that. I guess we need to view this as immersed submanifold somehow? Also, I will have to work with charts on spheres with seems rather painful. Any help will be appreciated!
 A: As I mentioned in my comment above, this is a special case of a more general result: if $F \to E \xrightarrow{p} B$ is a smooth fiber bundle, then the fibers of $\pi$ define a foliation of $E$ by leaves which are diffeomorphic to $F$. Let me first give the steps of how one proves this result. 


*

*The fibers of $p$ are embedded submanifolds of $E$ by the regular value theorem.

*By local triviality, we can find a collection of charts $\{(U_i, \eta_i)\}$ which cover $B$, and diffeomorphisms $\psi_i : p^{-1}(U_i) \to F\times U_i$ which respect the two natural projections to $U_i$.

*As $F$ is a manifold, we can find a collection of charts $\{(V_j, \chi_j)\}$ which cover $F$.

*The desired charts on $E$ are $\{(\psi_i^{-1}(V_j\times U_i), \phi_{i,j})\}$ where $\phi_{i,j} = (\chi_j, \eta_i)\circ\psi_i$.


I will carry out these steps (and verify the claims made) in the explicit case you asked about, namely the Hopf fibration.

Consider the map $p : S^3 \to \mathbb{CP}^1$ given by $p(z, w) = [z, w]$; note that the preimages under this map are precisely the equivalence classes: $p^{-1}(p(z, w)) = \{e^{i\theta}(z, w) \mid \theta \in \mathbb{R}\} = [(z, w)]$. The map $p$ is a surjective submersion, so every point in $\mathbb{CP}^1$ is a regular value and hence every preimage is an embedded submanifold of $S^3$. This establishes property $(1)$.
Consider the open set $U_1 = \mathbb{CP}^1\setminus\{[1, 0]\}$ and the map $\psi_1 : p^{-1}(U_1) \to S^1\times U_1$ given by $(z, w) \mapsto (\frac{w}{|w|}, [z, w])$. This is a smooth map with smooth inverse $(e^{i\theta}, [a, b]) \mapsto \frac{|b|e^{i\theta}}{b\sqrt{|a|^2+|b|^2}}(a, b)$. That is, $\psi_1$ is a diffeomorphism. Moreover, if we denote the natural projection $S^1\times U_1 \to U_1$ by $\operatorname{pr}_2$, then we have 
$$\operatorname{pr}_2(\Psi((z, w))) = \operatorname{pr}_2((\tfrac{w}{|w|}, [z, w])) = [z, w] = p((z, w)).$$ 
It follows that $[(z, w)]$ is mapped diffeomorphically to $S^1\times\{[z, w]\}$.
Note that the points $(i, [z, w]) \in S^1\times U_1$ are precisely those for which $\operatorname{arg}(w) = \frac{\pi}{2}$, so $\psi_1^{-1}((S^1\setminus\{i\})\times U_1) = p^{-1}(U_1)\setminus\{\operatorname{arg}(w) = \frac{\pi}{2}\}$. Each of the factors of $(S^1\setminus\{i\})\times U_1$ are diffeomorphic to Euclidean spaces. Explicitly, we have
diffeomorphisms $\eta_1 : U_1 \to \mathbb{R}^2$ given by $([z, w]) \mapsto (\operatorname{Re}(\frac{z}{w}), \operatorname{Im}(\frac{z}{w}))$ and $\chi_1 : S^1\setminus\{i\} \to \mathbb{R}$ given by $\lambda \mapsto \frac{\operatorname{Re}(\lambda)}{1-\operatorname{Im}(\lambda)}$; note that $\chi_1$ is precisely stereographic projection from the 'north pole'.
Putting what we have together, the map 
$$\phi_{1,1} : p^{-1}(U_1)\setminus\{\operatorname{arg}(w) = \tfrac{\pi}{2}\} \to \mathbb{R}^3$$ 
given by $\phi_{1,1} := (\chi_1, \eta_1)\circ\psi_1$ is a diffeomorphism and hence viewed as a coordinate chart. Moreover, if $L_{\alpha} = [(z, w)]$, then $(p^{-1}(U_1)\setminus\{\operatorname{arg}(w) = \frac{\pi}{2}\})\cap L_{\alpha}$ is mapped diffeomorphically to $\mathbb{R}\times\{(\operatorname{Re}(\frac{z}{w}), \operatorname{Im}(\frac{z}{w})\}$. That is, if $\varphi_{1,1} = (x^1, x^2, x^3)$, the intersection of the leaf with the coordinate chart is given by the equations $x^2 = \operatorname{Re}(\frac{z}{w}), x^3 = \operatorname{Im}(\frac{z}{w})$.
Replacing $\chi_1 : S^1\setminus\{i\} \to \mathbb{R}$ with $\chi_2 : S^1\setminus\{-i\}$ given by $\lambda \mapsto \frac{\operatorname{Re}(\lambda)}{1+\operatorname{Im}(\lambda)}$ (stereographic projection from the 'south pole'), we obtain a similar chart 
$$\phi_{1,2} = (\chi_2, \eta_1)\circ\psi_1 : p^{-1}(U_1)\setminus\{\operatorname{arg}(w) = \tfrac{3\pi}{2}\} \to \mathbb{R}^3.$$ 
Now let $U_2 = \mathbb{CP}^1\setminus\{[0, 1]\}$, then there is a diffeomorphism $\phi_2 : p^{-1}(U_2) \to S^1\times U_2$ respecting the two projections to $U_2$ given by $(z, w) \mapsto (\frac{z}{|z|}, [z, w])$. Replacing $\eta_1 : U_1 \to \mathbb{R}^2$ with $\eta_2 : U_2 \to \mathbb{R}^2$ given by $[z, w] \mapsto (\operatorname{Re}(\frac{w}{z}), \operatorname{Im}(\frac{w}{z}))$, we obtain two more charts 
$$\phi_{2,1} = (\chi_1, \eta_2)\circ\psi_2 : p^{-1}(U_2)\setminus\{\operatorname{arg}(w) = \tfrac{\pi}{2}\} \to \mathbb{R}^3$$ 
and 
$$\phi_{2,2} = (\chi_2, \eta_2)\circ\psi_2 : p^{-1}(U_2)\setminus\{\operatorname{arg}(w) = \tfrac{3\pi}{2}\} \to \mathbb{R}^3.$$
The domains of the four charts $\phi_{1,1}, \phi_{1, 2}, \phi_{2, 1}, \phi_{2,2}$ cover $S^3$ and satisfy the requirements of $(2)$.
