Abstract
We prove that if a topology on the real line endows it with a topological group structure (additive) for which the interval (0,+infty ) 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, so this topology is stronger than the usual topology, and as a consequence we obtain a characterization of the usual topology as lattice topology.
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