Topological group always means Hausdorff topological group, homomorphism (isomorphism) between topological groups always means continuous homomorphism (homeomorphic isomorphism). For a topological group G, the topological commutator subgroup (the closure of the algebraic commutator subgroup) is denoted by G’. For each locally compact group G, Takahashi has constructed a locally compact group GT (called the Takahashi quasi-dual) and a homomorphism G → GT such that GT is maximally almost periodic, and GT’ is compact. The category of all locally compact groups with these two properties is denoted by [TAK]. Takahashi's duality theorem states that G → GT is an isomorphism if G ∊ [TAK].