Q: Zariski projective Topology and quotient Topology A projective varieties in $\mathbb{P}^n$ is the common zero locus of some homogeneous polynomials.
It is simply to prove that the projective varieties on $\mathbb{P}^n$ are closed set of a Topology $\tau_z$, that we will define Zariski Topology on $\mathbb{P}^n$.
There is another natural Topology on $\mathbb{P}^n$ that is the quotient Topology $\tau_q$  induced by projection map $\pi : \mathbb{C}^{n+1}/ \{0\} \to \mathbb{P}^n$ where the Topology on $\mathbb{C}^{n+1}/ \{0\}$ is the Zariski Topology of $\mathbb{C}^{n+1}$ induced on the subset $\mathbb{C}^{n+1}/ \{0\}$.
It is correct? 
I want understand if there is a relationship between $\tau_z$ and $\tau_q$.
I think that they are equal because :


*

*$\pi$ is continuos if the Topology fixed on $\mathbb{P}^n$ is the Zariski Topology $\tau_z$ so $\tau_z\subseteq \tau_q$ ;

*If $X\subseteq \mathbb{P}^n$ is closed on $\tau_q$ then $\pi^{-1}(X)$ is closed on $\mathbb{C}^{n+1}/ \{0\}$ so there exists an ideal $I\subseteq \mathbb{K}[x_0,\dots , x_n]$ such that $\pi^{-1}(X)=Z(I)/ \{0\}$.
Then 
$X=\pi(\pi^{-1}(X))=\pi(Z(I)/ \{0\})=Z^{\mathbb{P}^n}(I)$ that is closed on $\tau_z$
