Let (
G, τ) be a commutative Hausdorff locally solid lattice group. In this paper we prove the following:
| (1) |
If (G, τ) has the A(iii)-property, then its completion
([^(G,)][^(t)] )(\widehat{G,}\widehat\tau )
is an order-complete locally solid lattice group.
|
| (2) |
If G is order-complete and τ has the Fatou property, then the order intervals of G are τ-complete.
|
| (3) |
If (G, τ) has the Fatou property, then G is order-dense in Ĝ and
([^(G,)][^(t)] )(\widehat{G,}\widehat\tau )
has the Fatou property.
|
| (4) |
The order-bound topology on any commutative lattice group is the finest locally solid topology on it.
|
As an application, a version of the Nikodym boundedness theorem for set functions with values in a class of locally solid
topological groups is established.
Keywords topological completion - locally solid ℓ-group - topological continuity - Fatou property - order-bound topology