R Hansen says: >If Neighbor understood real numbers and applied that to points, he would then be able to use what you wrote. He is still struggling with an incorrect sense of what points are. He still has a tendency (as all of us do) to think that two points can be beside each other, with no other point in between. The only way I know of getting over that tendency is to study the real numbers and what we mean by ?point? and ?line?.
You don't need to know everything about real numbers to follow my interpretation, and arrive at an understanding that (1) lines today are (usually) viewed as sets of points, (2) the cut point has to belong to one subset or the other but not both, assuming (3) you want exactly two subsets that are disjoint and when unioned give back the original.
That much is very elementary, naive, set theory. Sure there's that can be said, but one step at a time, huh? You don't need to get into density and limits and whatnot to look at what's going with the cut point itself.
I arrived at this by reading Neighbor's questions and trying honestly to see what he was after. I could be wrong.