On locally solid topological lattice groups

Khan, Abdul Rahim ; Rowlands, Keith

Czechoslovak Mathematical Journal, Tome 57 (2007) no. 3, pp. 963-973.

Résumé

Let $(G,\tau )$ be a commutative Hausdorff locally solid lattice group. In this paper we prove the following: (1) If $(G,\tau )$ has the $A$(iii)-property, then its completion $(\widehat{G},\hat{\tau })$ is an order-complete locally solid lattice group. (2) If $G$ is order-complete and $\tau $ has the Fatou property, then the order intervals of $G$ are $\tau $-complete. (3) If $(G,\tau )$ has the Fatou property, then $G$ is order-dense in $\widehat{G}$ and $(\widehat{G},\hat{\tau })$ 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.

Détail

Classification : 28B15, 46A40, 54H11
Keywords: topological completion; locally solid $\ell $-group; topological continuity; Fatou property; order-bound topology