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.
Yes (assuming $P$ is nonempty). If $P$ were reducible then it would have a pair of disjoint nonempty open subsets, meaning there would exist $V_1$ and $V_2$ open in $X$ such that $P\cap V_1$ and $P\cap V_2$ are nonempty but $P\cap V_1\cap V_2$ is empty. Now pick a point $p_i\in P\cap V_i$ and let $U_i$ be a compact open neighborhood of $p_i$ contained in $V_i$.