Embedded unbounded order convergent sequences in topologically convergent nets in vector lattices

Abstract

We show that, for a class of locally solid topologies on vector lattices, a topologically

convergent net has an embedded sequence that is unbounded order convergent to the

same limit. Our result implies, and often improves, many of the known results in this

vein in the literature. A study of metrisability and submetrisability of locally solid topologies on vector lattices is included.