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?

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?