In his book Differential and Integral Calculus, Edmund Landau gives an introductory chapter, and before starting it, he assumes as true some theorems. Among them, it is one he calls: "The deepest and most important of the fundan1ental properties of the real numbers"
Let there be given any division of all the real numbers into two classes, having the following properties:
$(a)$ Neither class is empty.
$(b)$ Every number of the first class is smaller than every number of the second class. (In other words, if $a<b$ and if $a$ lies in the second class, then $b$ lies in the second class.)
Then there exists a unique real number $\xi$ ; such that every $\eta < \xi$ belongs to the first class and every $\eta > \xi$ belongs to the second class.
What is this theorem called? Should it be a consequence of the least upper bound property of the real numbers?
If we change "real" by "rational" in the first sentence, this is the approach to Dedekind cuts and thus one of the possible constructions of the real numbers. The last sentence can be stated as "each of the division determines a one (two) unique set(s) of rational numbers" or something of the sort.