Let $X$ be a topological space. Set $K(X) := \{ A\subseteq X\mid A$ is quasi-compact and open $\}.$ A topological space $X$ is called spectral, if it satisfies all of the following conditions:
1) $X$ is quasi-compact and $T_0$. 2) $K(X)$ is a basis of open subsets of $X$. 3) $K(X)$ is closed under finite intersections. 4) $X$ is sober, i.e. every nonempty irreducible closed subset of $X$ has a (necessarily unique) generic point.
Let $C$ be a closed subset of a spectral topological space $X$. I am looking for equivalent conditions on $C$ under which if $A$ is a clopen(=Closed+Open) subset of $C$, then there exists a clopen subset $B$ of $X$ such that $A=C\cap B$?
