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Let $Y$ be a subset of a topological space $X$. We say that a clopen subset $L$ of $Y$ lifts to $X$ whenever there exists a clopen subset $H$ of $X$ such that $H\cap Y=L$.

Let $X$ be a compact and $T_0$-space and $Y$ be a closed subset of $X$. I am looking for conditions under which the clopen subsets of $Y$ lift to $X$. For example, if $Y$ is clopen then the clopen subsets of $Y$ lift to $X$.

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  • $\begingroup$ When $Y$ is clopen in $X$, there is no lifting required as $C \subset Y$ clopen in $Y$ then implies $C$ is clopen in $X$ too. If $Y$ is disconnected and $X$ is not, we cannot lift any clopens but $\emptyset$ and $Y$. $\endgroup$
    – Henno Brandsma
    Commented Jul 6, 2018 at 11:53

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