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!