Let $X$ be a non-empty spectral space and $P$ be a closed subset of $X$. If $U_1,U_2$ are two arbitary quasi-compact open subsets satisfying $P\cap U_1\neq\emptyset$ and $P\cap U_2\neq\emptyset$, then $P\cap U_1\cap U_2\neq\emptyset$.

Can we deduce that $P$ is irreducible from the above conditions?

Thanks in advance.