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 $X$.

  • $\begingroup$ 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 Jul 6 '18 at 11:53

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.