▁said ▁that ▁ ▁satisfies ▁Pon try agin ▁du ality ▁( or ▁that ▁ ▁is ▁a ▁ref lex ive ▁group , ▁or ▁a ▁reflect ive ▁group ). ▁This ▁has ▁been ▁extended ▁in ▁a ▁number ▁of ▁directions ▁beyond ▁the ▁case ▁that ▁ ▁is ▁locally ▁compact . ▁ ▁In ▁particular , ▁Samuel ▁Kap lan ▁showed ▁in ▁ 1 9 4 8 ▁and ▁ 1 9 5 0 ▁that ▁arbitrary ▁products ▁and ▁count able ▁inverse ▁limits ▁of ▁locally ▁compact ▁( H aus dor ff ) ▁ab elian ▁groups ▁satisfy ▁Pon try agin ▁du ality . ▁Note ▁that ▁an ▁infinite ▁product ▁of ▁locally ▁compact ▁non - comp act ▁spaces ▁is ▁not ▁locally ▁compact . ▁ ▁Later , ▁in ▁ 1 9 7 5 , ▁Rang ach ari ▁V enk at aram an ▁showed , ▁among ▁other ▁facts , ▁that ▁every ▁open ▁subgroup ▁of ▁an ▁ab elian ▁topological ▁group ▁which ▁satisfies ▁Pon try agin ▁du ality ▁itself ▁satisfies ▁Pon try agin ▁du ality . ▁ ▁More ▁recently , ▁Ser gio ▁Ar d anza - T rev ij ano ▁and ▁María ▁Jes ús ▁Ch as co ▁have ▁extended ▁the ▁results ▁of ▁Kap lan ▁mentioned ▁above . ▁They ▁showed ▁that ▁direct ▁and ▁inverse ▁limits ▁of ▁sequences ▁of ▁ab elian ▁groups ▁satisfying ▁Pon try agin ▁du ality ▁also ▁satisfy ▁Pon try agin ▁du ality ▁if ▁the ▁groups ▁are ▁met riz able ▁or ▁- spaces ▁but ▁not ▁necessarily ▁locally ▁compact , ▁provided ▁some ▁extra ▁conditions ▁are ▁satisfied ▁by ▁the ▁sequences . ▁ ▁However , ▁there ▁is ▁a ▁fundamental ▁aspect ▁that ▁changes ▁if ▁we