Topologies on the real line

Abstract

AbstractWe prove that if a topology on the real line endows it with a topological group structure (additive) for which the interval $$(0,+\infty )$$ ( 0 , + ∞ ) is an open set, so this topology is stronger than the usual topology. As a consequence we obtain characterizations of the usual topology as group topology and as ring topology. We also proved that if a topology on the real line is compatible with its usual lattice structure and is $$T_1$$ T 1 , so this topology is stronger than the usual topology, and as a consequence we obtain a characterization of the usual topology as lattice topology.