Some Analogues of Topological Groups

Abstract

Abstract Let (G, ∗) be a group and τ be a topology on G. Let τα = {A ⊆G : A ⊆ Int(Cl(Int(A)))}, g ∗ τ = {g ∗ A : A ∈ τ} for g ∈ G. In this paper, we establish two relations between G and τ under which it follows that g ∗ τ ⊆ τα and g ∗ τα ⊆ τα, designate them by α-topological groups and α-irresolute topological groups, respectively. We indicate that under what conditions an α-topological group is topological group. This paper also covers some general properties and characterizations of α-topological groups and α-irresolute topological groups. In particular, we prove that (1) the product of two α-topological groups is α-topological group, (2) if H is a subgroup of an α-irresolute topological group, then αInt(H) is also subgroup, and (3) if A is an α-open subset of an α-irresolute topological group, then < A > is also α−open. In the mid of discourse, we also mention about their relationships with some existing spaces.