> By the way I might be wrong of course, I'll be glad to have anyone > spot my error, my analogies might simply be misleading.
Why did Dedekind make his investigations?
Why did Bolzano feel compelled to prove the intermediate value theorem?
Why was Cauchy careful to not say that the fundamental sequences converged into the space from which their elements had been given?
I realize that you are not talking about those subjects. But you are taking them to the garbage heap -- along with every plausible piece of mathematics that uses the completeness axiom for the real numbers.
You cannot prove the fundamental theorem of algebra without results from analysis. It requires the existence of irrational roots for polynomials and the intermediate value theorem. So, you are tossing algebra onto the same heap with analysis.
Now, there is a circularity in the topology of real numbers. If you want to have
it must satisfy the axioms of a metric space. But those axioms are too strong.
Go get yourself a copy of "General Topology" by Kelley and read about uniformities and the metrization lemma for systems of relations.
What you will find is that the metric space axioms (the important direction associated with pseudometrics) depend on the least upper bound principle.
One can simply view it as fundamental sequences being grounded by cuts. It is not circular in that sense. It simply makes Dedekind prior to Cantor.
Before you continue with this mess, you should take some time to learn what it means for two real numbers to be equal to one another.