Definition:Limit Inferior of Net

Definition
Let $\left({S, \preceq}\right)$ be a directed set.

Let $L = \left({T, \precsim}\right)$ be a complete lattice.

Let $N: S \to T$ be a Moore-Smith sequence in $T$.

Then limes infimum of $N$ is defined as follows:
 * $\lim\inf N := \sup_L \left\{ {\inf_L\left({N\left[{\preceq\left({j}\right)}\right]}\right): j \in S}\right\}$

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