Definition:Limit Inferior of Net

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Definition


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In particular: Why is the preorder $\preceq$ and the order $\precsim$?
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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 $N$.


Also see

  • Results about limits inferior of nets can be found here.


Linguistic Note

The plural of limit inferior is limits inferior.

This is because limit is the noun and inferior is the adjective qualifying that noun.


Sources

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