Suppose we have a group $G$ acting on a topological space $X$, and there exists a $contractible$ open set $U$ in $X$ such that the orbit of $U$ is $X$.
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I'm not aware of a name for this. Any particular reason you're interested in it? – Qiaochu Yuan Oct 10 '19 at 07:53
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Algebraic topology. Maybe a better question would be what to call such a U or the cover provided by U. Maybe a "G-equivariant cover of X by contractible open sets"? – ZxJx Oct 10 '19 at 16:36