Suppose $X$ is a space, and $G$ a topological group acting continuously upon it. I'm interested in those closed sets $A \subseteq X$ whose translates under each $g \in G$, up to equivalence, are linearly ordered by set inclusion.
Question: Do these sets have a particular name, or have they been studied?
A quick example, suppose that $X = G = \mathbb{R}^2$, and the action is by addition. Then the collection of these sets is (I believe) precisely equal to the closed half-spaces.
To motivate, I am interested in continuous $G$-invariant weak orders on $X$; sets of the above form arise as the 'upper contour sets' of such orders.