Category:Definitions/Limits Inferior of Nets

This category contains definitions related to Limits Inferior of Nets.
Related results can be found in Category:Limits Inferior of Nets.

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 $\map \preceq j$ under $N$.