Suppose I have a covering map $\pi : E \to X$ between (nice) topological spaces, and $x \in X$. If $U \ni x$ is a very small open set, then $\pi^{-1}(U)$ is a discrete union of subsets $V\_d \subset X$ which are all homeomorphic to $U$ by $\pi$.



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I don't recall seeing a name for such sets. Perhaps the reason is that such individual sets $U$ are not so important, the existence of them is what is important. In fact if I considered naming such sets, I'd start wondering if there's some additional property of $U$ that's desirable (like, maybe you want the closure to also have a good lift). For the purpose of this question, I am fine with any such variant.


Specifically, I'd like to use such a name as an analog. I am in some kind of coarse geometry of groups context, specifically I have group $G$ with some generators $S$ and a quotient $G \to Q$, and I am interested in subsets $A \subset Q$ such that the preimage splits into $S$-disconnected chunks, one for each copy of $A$. I would've referred to these by the continuous analog, until I realized I don't know the name of such a thing. If you have a name suggestion for such $A$ directly, that works too.



Sets with a good lift under a covering
Such sets are frequently said to be 'evenly covered'.


