Predecessors and successors of the Euclidean topology on a subgroup of [formula omitted

Abstract

In this paper, we investigate the existence of predecessors and successors of the usual Euclidean topology τX on X in PG(X) and G(X), where X={(ab01):a>0,a,b∈R} and G(X) (PG(X)) is the lattice of all topological (paratopological) group topologies on X. We give a complete description of predecessors of τX in G(X) based on the fact that (X,τX) is a minimal Hausdorff topological group which was shown by Dierolf and Schwanengel in 1979. Then we give a negative answer to an open problem posed in [8]. Some constructions of successors of τX in PG(X) are also given. We also prove that τX has no successors in G(X).