Definition:Limit Inferior of Moore-Smith Sequence

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 limit inferior of $N$ is defined as follows:
 * $\liminf N := \sup_L \left\{ {\inf_L \left({N \left[{\preceq \left({j}\right)}\right]}\right): j \in S}\right\}$

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 $N$.