Let 
be a lattice (or a bounded lattice or a complemented lattice, etc.), and let 
be the covering relation of 
:
 |
Then 
is locally realized provided that for every finite subset 
of 
, there is a finitely generated sublattice 
of 
, that contains 
, for which 
. It can be shown that 
is locally realized if and only if there is a hyperfinitely
generated sublattice 
of 
such that 
. Using this characterization
of locally realized covering relations, the following standard result can be proved
using nonstandard methods:
Let 
be a locally finite lattice in which the covering relation is locally realized, and
let 
be the sublattice of 
which is generated by 
. Then 
is a connected tolerance of 
, and it is in fact the smallest locally subconnected (and
locally connected) tolerance of 
.
