Definition:Limit Inferior of Net

Definition
Let $\struct {S, \preceq}$ be a directed set.

Let $L = \struct {T, \precsim}$ be a complete lattice.

Let $N: S \to T$ be a net in $T$.

Then limit inferior of $N$ is defined as follows:
 * $\liminf N := \sup_L \set { {\map {\inf_L} {N \sqbrk {\map \preceq j} } : j \in S} }$

where
 * $\map \preceq j$ denotes the image of $j$ by $\preceq$,
 * $N \sqbrk {\map \preceq j}$ denotes the image of $\preceq \left({j}\right)$ under $N$.