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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?

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    $\begingroup$ Should be "if and only if every non-empty subset of $X$ which has an upper bound has a supremum in $X$". $\endgroup$ – Bungo Nov 10 '15 at 23:27
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    $\begingroup$ It should indeed say "every non-empty bounded subset of $X$ has a supremum in $X$." $\endgroup$ – Noah Schweber Nov 10 '15 at 23:27
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    $\begingroup$ And I have no idea why someone downvoted this intelligent question. $\endgroup$ – Brian M. Scott Nov 10 '15 at 23:28
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    $\begingroup$ For anyone who goes to look, I've now fixed the Wikipedia article! $\endgroup$ – Sharkos Nov 10 '15 at 23:30
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    $\begingroup$ @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?". $\endgroup$ – Asaf Karagila Nov 10 '15 at 23:39
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(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 $X$.

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    $\begingroup$ @EsperantoSpeaker1 maybe this is a language issue, but I find your tone excessive. There was a minor error, basically a typo, textbooks and research articles are full of such things. Besides, it seems all but one version were correct. With some effort on your side, you could just have corrected it there. $\endgroup$ – quid Nov 11 '15 at 10:56
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    $\begingroup$ @quid: So, engagement with Wikipedia is compact? That is, if you detect a bug you are required to fix it - the middle ground of merely reporting it is not an option - or even of getting a second opinion before doing so? Given Asaf Karagila who says the OP should take the “What have I misunderstood in...” stance, and you who say the OP should just charge on ahead, I would say that the solution lies on the imaginary axis. $\endgroup$ – user27325 Nov 11 '15 at 22:02
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    $\begingroup$ @EsperantoSpeaker1 no, you can report a bug, but you should report it where it occurs if possible. There is infrastucture at Wikipedia for just that. Also for asking for clarification. It is ultimately quite harmful if the infrastructure that a project provides for addressing its issues is not used but quasi-random other channels are used. $\endgroup$ – quid Nov 11 '15 at 22:08
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    $\begingroup$ @EsperantoSpeaker1 further on the perceived discrpancy between what Asaf and I say. Ultimately we both are not overly thrilled about your approach and tone related to the matter: "glaring error," "dismayed" etc. All that because somebody dropped a word when moving around some text a couple days ago. $\endgroup$ – quid Nov 11 '15 at 22:11
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    $\begingroup$ @EsperantoSpeaker1 I take it you mean to chose your words more carefully next time. That's great. Anyway, I think we discussed this enough. Your question was not that bad, either. In some sense I made too much fuss about it. $\endgroup$ – quid Nov 12 '15 at 1:02

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