Separability of Topological Groups: A Survey with Open Problems

Abstract

Separability is one of the basic topological properties. Most classical topological groups and Banach spaces are separable; as examples we mention compact metric groups, matrix groups, connected (finite-dimensional) Lie groups; and the Banach spaces C ( K ) for metrizable compact spaces K; and ℓ p , for p ≥ 1 . This survey focuses on the wealth of results that have appeared in recent years about separable topological groups. In this paper, the property of separability of topological groups is examined in the context of taking subgroups, finite or infinite products, and quotient homomorphisms. The open problem of Banach and Mazur, known as the Separable Quotient Problem for Banach spaces, asks whether every Banach space has a quotient space which is a separable Banach space. This paper records substantial results on the analogous problem for topological groups. Twenty open problems are included in the survey.

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.…”