In A. Bondal, M. van den Bergh's paper, **Generators and representability of functors in commutative and noncommutative geometry** , the "reduction principle" of quasi-compact, quasi-separated schemes is shown. It is stated as follows.


> Assume $X = U_1 \cup U_2 $ with $U_1, U_2$ open and put $U_{12} = U_1 \cap U_2$.

> Let $P$ be a property satisfied by some schemes such that
> 
> (1) $P$ is true for affine schemes.
> 
> (2) If $P$ holds for $U_1, U_2, U_{12}$ as above, then it holds for $X$. 
> 
> Then $P$ holds for all quasi-compact quasi-separated schemes.

I understood this statement. However, the next **Remark3.3.2** says

>It is easy to see that the class of quasi-compact quasi-separated schemes is the biggest class of schemes to which the reduction principle is applicable (for all properties $P$).

 I'm not sure why this characterization of quasi-compact, quasi-separated schemes holds. Is there an obvious property $P$ which satisfies above properties and holds only for quasi-compact, quasi-separated schemes?