The Tubby Torus as a Quotient Group

Abstract

Let E be any metrizable nuclear locally convex space and E ^ the Pontryagin dual group of E. Then the topological group E ^ has the tubby torus (that is, the countably infinite product of copies of the circle group) as a quotient group if and only if E does not have the weak topology. This extends results in the literature related to the Banach–Mazur separable quotient problem.

Quotient on some Generalizations of topological group

“…In addition, the structures were reformed algebraically [31][32][33][34] and topologically 35 via fuzzy metrics [36][37][38][39] . Quotient forms such as factorizable [40][41][42] , metrizable 43,44 , and separable [45][46][47] especially in Banach spaces [48][49][50][51] were developed recently. Bounded groups 52,53 , Matrix groups 54 , homotopy characteristics [55][56][57][58], and projectivity 59 on topological groups were remarkable.…”