I heard the following two questions recently from Carl Mummert, who encouraged me to spread them around. Part of his motivation for the questions was to give the subject of computable model theory some traction on complete metric spaces, by considering the countable objects as stand-ins for the full spaces, to the extent that they are able to do so.
Question 1. Is there a countable subset D of the real plane $R^2$ that is dense and has the property that the distance d(x,y) is a rational number for all $x,y\in D$?
The one dimensional analogue of this question has an easy affirmative answer, since the rationals Q sits densely in R and the distance between any two rationals is rational.
Question 2. More generally, does every separable complete metric space have a countable dense set D with all distances between elements of D rational? [Edit: Tom Leinster has pointed out that if the space has only two points, at irrational distance, this fails. So let us consider the case of connected spaces, generalizing the situation of Question 1.]
If one is willing to change to an equivalent metric (giving rise to the same topology), then the answer to Question 1 is Yes, since the rational plane Q x Q is dense in the real plane R x R, and has all rational distances under the Manhattan metric, which gives rise to the same topology. Is the answer to the correspondingly weakened version of Question 2 also affirmative, if one is willing to change the metric?
Note that one cannot find an equivalent metric such that all distances in $R^2$ become rational, since omitted values in the distance function lead to disconnectivity in the space. This is why the questions only seek to find a dense subset with the rational condition.
The question seems related to the question of whether it is possible to find large non-linear arrangements of points in the plane with all pair-wise distances being integers. For example, this is true of the integers Z sitting inside R, but can one find a 2 dimensional analogue of this? Clearly, some small arrangements (triangles, etc.) are possible, but I am given to understand that there is a finite upper bound on the size of such arrangements. What is the precise statement about this that is known?