How to understand $\limsup$/$\liminf$ of a subset of a complete lattice with a topology

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