The axiom is given in this picture:=
I actually have lot of questions from this topic but I post only one here: What does axiom 6 actually mean? Can only one real number lie in a interval $[a(S), a(T)]$?
Also can only one number lie in $(a(S), a(T))$ ?
Infinitely many real numbers lie in an interval such as $[a(S), a(T)]$ for a given $S$ and $T$, but that is not what the axiom is referring to. Note that the axiom says "... for all step regions $S$ and $T$ satisfying (1.1)..." So there are infinitely many $S$ and $T$ to consider. Only one real number lies in all those infinitely many intervals.
As an example, consider the intervals
$$\left[1-\frac 1n, 1+\frac 1n\right]$$
where $n$ is a positive integer. Then for any particular value of $n$ there are infinitely many real numbers in that interval. However, only one real number, namely $1$, lies in all of those intervals.