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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 $T$

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  • $\begingroup$ What about the case of finite $T_0$-spaces? Is everything clear in this case? $\endgroup$ – Taras Banakh Jun 30 '18 at 22:07

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