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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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    $\begingroup$ :) $\endgroup$ – t.b. Nov 24 '11 at 2:31
  • $\begingroup$ @t.b.: :-D. Yes, I haven't forgot it. $\endgroup$ – Tim Nov 24 '11 at 2:34
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    $\begingroup$ @Tim: Wikipedia is made by you! I hope you can go there and improve it... ;-) $\endgroup$ – André Caldas Nov 24 '11 at 2:46
  • $\begingroup$ @AndréCaldas: Yes, and Wiki is my teacher! $\endgroup$ – Tim Nov 24 '11 at 2:47
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    $\begingroup$ 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 $\mathbb R$. $\endgroup$ – Henning Makholm Nov 24 '11 at 3:03

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