Dedekind cut
It is straightforward to show that a Dedekind cut among the real numbers is uniquely defined by the corresponding cut among the rational numbers. Similarly, every cut of reals is identical to the cut produced by a specific real number (which can be identified as the smallest element of the "B" set). In other words, the number line where every real number is defined as a Dedekind cut of rationals is a complete continuum without any further gaps.