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From Wikipedia:

Let $Y$ be a partially ordered set which is also a topological space and a complete lattice so that the suprema and infima always exist. For a set $X ⊆ Y$, define $$ \liminf X := \inf \{ x \in Y : x \text{ is a limit point of } X \}\, $$ $$ \limsup X := \sup \{ x \in Y : x \text{ is a limit point of } X \}\, $$

I wonder if this definition is consistent in some way with the informal interpretation of $\limsup$ as $\inf \sup$, and of $\liminf$ as $\sup \inf$, or just another case that you "don't know why Wikipedia writes what it writes"? Thanks!

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    [:)](http://chat.stackexchange.com/transcript/36?m=2505733#2505733)2011-11-24
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    @t.b.: :-D. Yes, I haven't forgot it.2011-11-24
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    @Tim: Wikipedia is made by you! I hope you can go there and improve it... ;-)2011-11-24
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    @AndréCaldas: Yes, and Wiki is my teacher!2011-11-24
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    I guess it is not right... it seems the article is trying to generalize the concept for sets, instead of sequences or nets. I guess the author is violating the Wikipedia guidelines. Wikipedia is not a place for development of new theories. It is a place for consolidated stuff. This article lacks a lot of references to what it says. There is a standard way to flag an entry as "lacking references". I think you should go there and improve it... at least "flag" the lack of references... :-) http://en.wikipedia.org/wiki/Wikipedia:No_original_research2011-11-24
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    How to flag lack of references: http://en.wikipedia.org/wiki/Template:Citations_missing2011-11-24
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    I don't think we need to assume that it is just something an editor personally made up, but I agree with André that it doesn't look like a standard notion. One thing that seems particularly nonstandard is that it does not seem to presuppose _any_ particular relation between the topology (used to identify limit points) and the lattice structure (used to take infima and supreme). There are several non-equivalent ways to equip a complete lattice with a topology ...2011-11-24
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    Also, the grafted-on note that "neither the limit inferior nor the limit superior of a set must be an element of the set" simply cannot be right. Nothing even slightly like it is true for the standard concepts in $\mathbb R$.2011-11-24
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    @HenningMakholm: Thanks! I think the quote says when $X$ is a subset of the space $Y$, $\liminf X$ and $\limsup X$ as defined there may not be in $X$, although they must be in $Y$. So I don't see it is wrong.2011-11-24
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    @Tim, but for real sequences it is perfectly allowed for the liminf of a sequence to be in the sequence itself. Also, if you apply the displayed _definition_ of limsup to a closed interval $X$ ($\subseteq \mathbb R$ with its usual order and topology), you get its upper endpoint, which is a member of $X$.2011-11-24
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    @Henning: I think this is a misunderstanding -- the Wikipedia text is saying "neither must be an element" in the sense of "neither need be an element".2011-11-24
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    @joriki, that makes sense. Edited.2011-11-24

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