# Is Wikipedia wrong about the least-upper-bound property?

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?

• Should be "if and only if every non-empty subset of $X$ which has an upper bound has a supremum in $X$". – Bungo Nov 10 '15 at 23:27
• It should indeed say "every non-empty bounded subset of $X$ has a supremum in $X$." – Noah Schweber Nov 10 '15 at 23:27
• And I have no idea why someone downvoted this intelligent question. – Brian M. Scott Nov 10 '15 at 23:28
• For anyone who goes to look, I've now fixed the Wikipedia article! – Sharkos Nov 10 '15 at 23:30
• @Brian: We're not in disagreement here. I'm just saying why I would have downvoted something whose title is "Is X wrong about Y?". – Asaf Karagila Nov 10 '15 at 23:39

(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 $X$.