My actual question is related to an unusual circumstance in a game which I am playing. We are trying to move one point-like object from one part of the moon to another without passing through another player's land. However, due to a loophole, nothing prevented one friend from buying a dense set of measure zero. (The game encourages rules-lawyering so this isn't completely out of line). We now don't know what's going on.
I imagine the simplest version of this question would be if there is a continuous nontrivial curve in $\mathbb{R}^2$ that avoid every point in $\mathbb{Q}^2$, but now I'm curious if there's a way we that this statement could be generalized to arbitrary topological spaces (without necessarily having a measure).
Technically, according to this game the object has to have a velocity, so to resolve the confusion for us we'd need a curve that is also non-differentiable at at most countably many points, but I don't know enough about differentiablity to imagine the appropriate generalization.