Regarding infinite sets, I simply meant that defining the problem in terms of floor() was problematic. You avoided that in your reply by using the phrases "two parts of equal size" and "two equal parts".

In my original statement of the problem, I was describing, finite, discrete sets. After all, I was saying one set had N points, the other M points. I would not be able to give a number of points if any of the sets was infinite. In my later reply, I was refering to the continuous form of the problem, in which we have continuous sets (but not necessarely connected). Then it's right to talk about two parts of equal size.

Abigail