A dense point set, no three points on a line

The other day I was thinking about sets of points in general position, which means that no three of the points are on a single straight line. Can you think of an infinite such set?

One answer: Here's one way to do it: just take the set of points (x,y) with x a positive integer and y=x². Then you can see by the convexity of the parabola that no three of these points are in a line.

So that's an infinite set. Can you think of a set X in general position which is also dense, meaning that every point in the plane is a limit of some subsequence of points from X?

The solution is actually kind of easy (where by "easy" I mean that math professors would call it easy :) Check it out: [PDF] [TEX].