Generic properties of topological groups (original) (raw)

A Note in the Topological Groups

2015

In this note, for a topological group , we introduce a new concept as bounded topological group, that is, is called bounded, if for every neighborhood of identity element of , there is a natural number such that . We study some properties of this new concept and its relationships with other topological properties of topological groups.

Generic countably infinite groups

arXiv (Cornell University), 2021

Countably infinite groups (with a fixed underlying set) constitute a Polish space G with a suitable metric, hence the Baire category theorem holds in G. We study isomorphism invariant subsets of G, which we call group properties. We say that the generic countably infinite group is of property P if P is comeager in G. We prove that every group property with the Baire property is either meager or comeager. We show that there is a comeager elementary equivalence class in G but every isomorphism class is meager. We prove that the generic group is algebraically closed, simple, not finitely generated and not locally finite. We show that in the subspace of Abelian groups the generic group is isomorphic to the unique countable, divisible torsion group that contains every finite Abelian group. We sketch the model-theoretic setting in which many of our results can be generalized. We briefly discuss a connection with infinite games.

Some Baire category properties of topological groups

arXiv (Cornell University), 2019

We present several known and new results on the Baire category properties in topological groups. In particular, we prove that a Baire topological group X is metrizable if and only if X is point-cosmic if and only if X is a σ-space. A topological group X is Choquet if and only if its Raikov completionX is Choquet and X is G δ-dense inX. A topological group X is complete-metrizable if and only if X is a point-cosmic Choquet space if and only if X is a Choquet σ-space. Finally, we pose several open problem, in particular, whether each Choquet topological group is strong Choquet.

Categorically compact topological groups

Journal of Pure and Applied Algebra, 1998

We study the notion of a categorically compact topological group, suggested by the Kuratowski-Mrowka characterization of compact spaces. A topological group G is categorically compact, or C-compact, if for any topological group H the projection G x H-+ H sends closed subgroups to closed subgroups. We prove, among others, the following theorems: (1) any product of Ccompact topological groups is C-compact; (2) separable C-compact groups are totally minimal; (3) C-compact soluble topological groups are compact. 01998 Elsevier Science B.V.

On generalized topological groups

In this work, we will introduce the notion of generalized topological groups using generalized topological structure and generalized continuity defined byÁ. Császár [2]. We will discuss some basic properties of this kind of structures and connectedness properties of this structures are given.

Separability of Topological Groups: A Survey with Open Problems

Axioms

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.

The separable quotient problem for topological groups

Israel Journal of Mathematics

The famous Banach-Mazur problem, which asks if every infinitedimensional Banach space has an infinite-dimensional separable quotient Banach space, has remained unsolved for 85 years, though it has been answered in the affirmative for reflexive Banach spaces and even Banach spaces which are duals. The analogous problem for locally convex spaces has been answered in the negative, but has been shown to be true for large classes of locally convex spaces including all non-normable Fréchet spaces. In this paper the analogous problem for topological groups is investigated. Indeed there are four natural analogues: Does every non-totally disconnected topological group have a separable quotient group which is (i) non-trivial; (ii) infinite; (iii) metrizable; (iv) infinite metrizable. All four questions are answered here in the negative. However, positive answers are proved for important classes of topological groups including (a) all compact groups; (b) all locally compact abelian groups; (c) all σ-compact locally compact groups; (d) all abelian pro-Lie groups; (e) all σ-compact pro-Lie groups; (f) all pseudocompact groups. Negative answers are proved for precompact groups.

On topological groups with a small base and metrizability

Fundamenta Mathematicae, 2015

A (Hausdorff) topological group is said to have a G-base if it admits a base of neighbourhoods of the unit, {Uα : α ∈ N N }, such that Uα ⊂ U β whenever β ≤ α for all α, β ∈ N N. The class of all metrizable topological groups is a proper subclass of the class TG G of all topological groups having a G-base. We prove that a topological group is metrizable iff it is Fréchet-Urysohn and has a G-base. We also show that any precompact set in a topological group G ∈ TG G is metrizable, and hence G is strictly angelic. We deduce from this result that an almost metrizable group is metrizable iff it has a G-base. Characterizations of metrizability of topological vector spaces, in particular of Cc(X), are given using G-bases. We prove that if X is a submetrizable kω-space, then the free abelian topological group A(X) and the free locally convex topological space L(X) have a G-base. Another class TGCR of topological groups with a compact resolution swallowing compact sets appears naturally. We show that TGCR and TG G are in some sense dual to each other. We conclude with a dozen open questions and various (counter)examples.