Abstract
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 (Ǧ, τ̌) 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 (Ǧ, τ̌) 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.
| Original language | English |
|---|---|
| Pages (from-to) | 963-973 |
| Number of pages | 11 |
| Journal | Czechoslovak Mathematical Journal |
| Volume | 57 |
| Issue number | 3 |
| DOIs | |
| State | Published - Sep 2007 |
Bibliographical note
Funding Information:Clearly, |η| is bounded on order-bounded sets; also |η|(a) = η∗(a) for all a > 0. This implies that |η|(|a|) = |η|(a) (a ∈ G), and so the topology τb′ defined by the quasi-norms {|η|} (η ∈ S) is locally solid. Thus by the first part of the proof τb′ ⊆ τb. On the other hand, if U is any τb-neighborhood of 0 in G, then there exists a q ∈ S and a positive number ε such that {x: q(x) < ε} ⊆ U. Now q(x) ⩽ 2|q|(x) ([15], Lemma 3), and so we have {x: |q|(x) < ε/2} ⊆ U. This implies that τb ⊆ τb′ and so τb = τb′. Thus τb is locally solid, as required. □ Acknowledgment. The author A. R. Khan gratefully acknowledges support provided by the King Fahd University of Petroleum & Minerals during this research.
Keywords
- Fatou property
- Locally solid l-group
- Order-bound topology
- Topological completion
- Topological continuity
ASJC Scopus subject areas
- General Mathematics