how do I show that Segre embedding is a homeomorphism? I have to check that the diagonal of space projective $ \mathbb P^n$ is closed in the product $ \mathbb P^n$x$ \mathbb P^n$. So I need to show that Segre embedding is a homeomorphism. How do I do this?  
the projective spaces over k, with k a algebraically closed field. And I using the Zariski topology. 
Specifically my doubt is Proposition 7.11(b) and Corollary 7.13 of Gathmann. Link below: pdf

A counterexample comes from $\Psi(\mathbb{P}^1 \times \mathbb{P}^1) \cong Q= V(xw-yz) \subset \mathbb{P}^3$, as $Q$ contains curves (such as the 3-uple embedding of $\mathbb{P}^1$ in $\mathbb{P}^3$, $\nu_3(\mathbb{P}^1) \cong V(xz-y^2, xw-yz, yw-z^2)$ which do not correspond to a curve in $\mathbb{P}^1 \times \mathbb{P}^1$ in the product topology.
That is, the Segre embedding $\Psi(\mathbb{P}^n \times \mathbb{P}^m) \hookrightarrow \mathbb{P}^{nm + n + m}$ is not a homeomorphism onto its image in general if we put the product topology on $\mathbb{P}^n\times \mathbb{P}^m$. This raises the question of what topology you’re putting on the domain $\mathbb{P}^n \times \mathbb{P}^m$.
