Wikipedia states that the least-upper-bound property “ is a fundamental property of the real numbers and certain other ordered sets. A set $X$ has the least-upper-bound property if and only if every non-empty subset of $X$ has a supremum in $X$.”
This seems to me to be a bad slip, because from that it follows that $\mathbb R$ does not have the least-upper-bound property. (Later in the article Wikipedia gives what I would consider to be the correct formulation of the property.)
So, am I missing something, or is this a glaring error in Wikipedia?
(As stated by Noah above), another necessary condition is that every such subset $X$ be bounded from above. That is, every non-empty bounded subset of $X$ (be it closed, open, or neither) has a supremum in $X$.