Abstract
Let (G, T ) be a topological group and let Tα denote the topology generated by the α-open subsets of (G, T ). A subset A of a topological space is called α-open if A ⊆ Int cl(Int(A)) . The notion of α-open set was introduced by Njastad as one of the fundamental classes of nearly open sets. The purpose of this paper is to investigate Polish topological groups from the point of view of the α-topology. We study the relation between T and Tα, the corresponding α-Borel structure, translations, inversion, group homomorphisms, subgroups, products, completions, and separability. We prove several criteria under which the α-topology preserves the Polish character of a topological group. In particular, we isolate the exact obstruction: although Tα is always a topology containing T , it need not automatically be a Polish group topology. We therefore formulate intrinsic conditions in terms of countable α-bases, compatible complete metrics, and continuity of the group operations. We also establish sufficient conditions for equality of the ordinary and α-Borel structures. Examples are given to distinguish the ordinary Polish topology from the finer α-topology.