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.
Highlights
All topological spaces and topological groups are assumed to be Hausdorff and all topological spaces are assumed to be infinite unless explicitly stated otherwise.The fundamental topological operations which produce new topological groups from given ones are: (1) (2) (3) (4)taking subgroups; taking finite or infinite products; open continuous homomorphic images = quotient images;(topological group) isomorphic embeddings.A topological space which has a dense countable subspace is called separable.The main aim of this survey paper is to present systematically the results concerning the behavior of separability of topological groups with respect to the topological operations listed above and make clear which problems are open
Much of the material is from the recent publications [1,2,3,4,5,6,7,8]. Speaking, this survey contributes to the manifestation of the phenomenon that the structure of topological groups is much more sensitive to the presence of countable topological properties than is the structure of general topological spaces
In the result, which complements Theorem 5, we identify a large class of topological groups with the class of closed subgroups of separable path-connected, locally path-connected topological groups
Summary
All topological spaces and topological groups are assumed to be Hausdorff and all topological spaces are assumed to be infinite unless explicitly stated otherwise. The main aim of this survey paper is to present systematically the results concerning the behavior of separability of topological groups with respect to the topological operations listed above and make clear which problems are open. Much of the material is from the recent publications [1,2,3,4,5,6,7,8] Speaking, this survey contributes to the manifestation of the phenomenon that the structure of topological groups is much more sensitive to the presence of countable topological properties than is the structure of general topological spaces. The reader is advised to consult the monographs of Engelking [9] and Arhangel’skii and Tkachenko [10] for any notions which are not explicitly defined in our paper
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