Archimedean property application

Archimedean property application

The Archimedean property states that
for all $a \in \Bbb R$ and for some $n \in \Bbb N: a < n$.
Similarly,
for all $n \in \Bbb N | b \in \Bbb R | 0 < {1 \over n} < b$.
Thus, there can be found a rational number such that it is the boundary
point of some open set.
In other words, there can be made Dedekind cuts $(A, B)$.
Is it true we can make infinitely many open sets - that is, does this
imply that there can be infinitely many disjoint open sets $A, B$ - all
bounded by some Dedekind cut?