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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.

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    @anonymous: It's a game of http://en.wikipedia.org/wiki/Nomic. I'm actually not the record keeper for our game, so if you'd like to see the ruleset I could host it somewhere but you'd have to be patient. Also Nomic's infobox on wiki is hilarious. "Players: variable. Setup time: variable… Random Chance: variable. Skill(s) required: …variable."2012-12-09

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To answer your simple version, the line $y=\pi x+\sqrt{2}$ avoids every point in $\mathbb{Q}^2$ since $\sqrt{2}$ is algebraic and $\pi$ is not.

There are dense sets of measure zero which prevent this sort of approach such as $ (\mathbb{R}\times \mathbb{Q}) \cup (\mathbb{Q}\times \mathbb{R})$ so it rather depends on precisely what set your friend bought.

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    I believe the wording was a particular projection of $\mathbb{Q^2}$ onto the surface of the moon, which we are essentially assuming is spherical.2012-12-09