Huh? Why can't the dividing point be unique if it's in one of the classes. It's the point that "produces the division" it is not the point that is between both classes. Like I said (I think), "produces the division" is not a technical term, but it's meaning is obvious. The other thing to note that while sup((-infinity, 0)) = 0 = inf([0, infinity)), 0 is an element of [0, infinity) while not an element of (-infinity, 0). That is, the supremum, if it exists, of an open interval of the reals is not in the interval. This is how the supremum of one class is the infimum of the other, without this unique point being in both classes.