Let $A$ be a subset of a topological space $T$, we say that clopen subset of $A$ lift to $T$ whenever $L$ is a clopen subset of $A$ the there exists a clopen subset $H$ of $T$ such that $H\cap A=L$.
Now let $X$ be a compact and $T_0$-space and $Y$ and $Z$ be two closed subsets of $X$ whose clopen subsets lift to $X$. Is there any condition under which the clopen subsets of $Y\cap Z$ lift to $T$? and is there any condition under which the clopen subsets of $Y\cup Z $ lift to $T$
