I think that your posting shows a good deal of confusion.
I have taught courses in which the DEDEKIND CUTS have been central to the basic introduction to the structure of the real numbers. However, what you have posted has no relation to any notation about “CUTS” that I have ever seen.
Let me give you a standard (and elementary) reference for this topic.
Look in your mathematics library for PRICIPLES OF MATHEMATICAL ANALYSIS by Walter Rudin.