I've just started on Principles of Mathematical Analysis by Walter Rudin (Third Edition) and came across the following definition for the least-upper-bound property:
Definition: An ordered set $S$ is said to have the least-upper-bound property if the following is true:
If $E\subset S$, $E$ not empty, and $E$ is bounded above, then sup $E$ exists in $S$.
My question is the following: Does this mean that open bounded sets of the form $(a,b)$ do not have the least-upper-bound property, since the supremum of the subset $(\frac{a+b}{2},b)$ is $b\notin (a,b)$ ?