Products of topological groups in which all closed subgroups are separable (original) (raw)

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.

Forcing hereditarily separable compact-like group topologies on Abelian groups

Topology and its Applications, 2005

Let c denote the cardinality of the continuum. Using forcing we produce a model of ZFC + CH with 2 c "arbitrarily large" and, in this model, obtain a characterization of the Abelian groups G (necessarily of size at most 2 c) which admit: (i) a hereditarily separable group topology, (ii) a group topology making G into an S-space, (iii) a hereditarily separable group topology that is either precompact, or pseudocompact, or countably compact (and which can be made to contain no infinite compact subsets), (iv) a group topology making G into an S-space that is either precompact, or pseudocompact, or countably compact (and which also can be made without infinite compact subsets if necessary).

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.

Density character of subgroups of topological groups

Transactions of the American Mathematical Society, 2015

It is well-known that a subspace Y of a separable metrizable space X is separable, but without X being assumed metrizable this is not true even in the case that Y is a closed linear subspace of a topological vector space X. Early this century K.H. Hofmann and S.A. Morris introduced the class of pro-Lie groups which consists of projective limits of finite-dimensional Lie groups and proved that it contains all compact groups, all locally compact abelian groups, and all connected locally compact groups and is closed under the formation of products and closed subgroups. They defined a topological group G to be almost connected if the quotient group of G by the connected component of its identity is compact. We prove that an almost connected pro-Lie group is separable if and only if its weight is not greater than the cardinality c of the continuum. It is deduced from this that an almost connected pro-Lie group is separable if and only if it is homeomorphic to a subspace of a separable Hausdorff space. It is also proved that a locally compact (even feathered) topological group G which is a subgroup of a separable Hausdorff topological group is separable, but the conclusion is false if it is assumed only that G is homeomorphic to a subspace of a separable Tychonoff space. It is shown that every precompact (abelian) topological group of weight less than or equal to c is topologically isomorphic to a closed subgroup of a separable pseudocompact (abelian) group of weight c. This result implies that there is a wealth of closed non-separable subgroups of separable pseudocompact groups. An example is also presented under the Continuum Hypothesis of a separable countably compact abelian group which contains a non-separable closed subgroup. It is proved that the following conditions are equivalent for an ω-narrow topological group G: (i) G is homeomorphic to a subspace of a separable regular space; (ii) G is a subgroup of a separable topological group; (iii) G is a closed subgroup of a separable path-connected locally path-connected topological group.

CLP-compactness for topological spaces and groups

Topology and its Applications, 2007

We study CLP-compact spaces (every cover consisting of clopen sets has a finite subcover) and CLP-compact topological groups. In particular, we extend a theorem on CLP-compactness of products from [J. Steprāns, A. Šostak, Restricted compactness properties and their preservation under products, Topology Appl. 101 (3) (2000) 213-229] and we offer various criteria for CLP-compactness for spaces and topological groups, that work particularly well for precompact groups. This allows us to show that arbitrary products of CLP-compact pseudocompact groups are CLP-compact. For every natural n we construct: (i) a totally disconnected, n-dimensional, pseudocompact CLP-compact group; and (ii) a hereditarily disconnected, n-dimensional, totally minimal, CLP-compact group that can be chosen to be either separable metrizable or pseudocompact (a Hausdorff group G is totally minimal when all continuous surjective homomorphisms G → H , with a Hausdorff group H , are open).

Abelian groups admitting a Fréchet–Urysohn pseudocompact topological group topology

Journal of Pure and Applied Algebra, 2010

We show that every Abelian group G with r 0 (G) = |G| = |G| ω admits a pseudocompact Hausdorff topological group topology T such that the space (G, T) is Fréchet-Urysohn. We also show that a bounded torsion Abelian group G of exponent n admits a pseudocompact Hausdorff topological group topology making G a Fréchet-Urysohn space if for every prime divisor p of n and every integer k ≥ 0, the Ulm-Kaplansky invariant f p,k of G satisfies (f p,k) ω = f p,k provided that f p,k is infinite and f p,k > f p,i for each i > k. Our approach is based on an appropriate dense embedding of a group G into a Σproduct of circle groups or finite cyclic groups.

Topological groups and convex sets homeomorphic to non-separable Hilbert spaces

Central European Journal of Mathematics, 2008

Let X be a topological group or a convex set in a linear metric space. We prove that X is homeomorphic to (a manifold modeled on) an infinite-dimensional Hilbert space if and only if X is a compltetely metrizable absolute (neighborhood) retract with ω-LFAP, the countable locally finite approximation property. The latter means that for any open cover U of X there is a sequence of maps (fn : X → X)n∈ω such that each fn is U-near to the identity map of X and the family {fn(X)}n∈ω is locally finite in X. Also we show that a metrizable space X of density dens(X) < d is a Hilbert manifold if X has ω-LFAP and each closed subset A ⊂ X of density dens(A) < dens(X) is a Z∞-set in X.

Bounded sets in spaces and topological groups

Topology and its Applications, 2000

We investigate C-compact and relatively pseudocompact subsets of Tychonoff spaces with a special emphasis given to subsets of topological groups. It is shown that a relatively pseudocompact subset of a space X is C-compact in X, but not vice versa. If, however, X is a topological group, then these properties coincide. A product of two C-compact (relatively pseudocompact) subsets A of X and B of Y need not be C-compact (relatively pseudocompact) in X ×Y , but if one of the factors X, Y is a topological group, then both C-compactness and relative pseudocompactness are preserved. We prove under the same assumption that, with A and B being bounded subsets of X and Y , the closure of A × B in υ(X × Y ) is naturally homeomorphic to cl υX A × cl υY B, where υ stands for the Hewitt realcompactification. One of our main technical tools is the notion of an R-factorizable group. We show that an R-factorizable subgroup H of an arbitrary group G is z-embedded in G. This fact is applied to prove that the group operations of an R-factorizable group G can always be extended to the realcompactification υG of G, thus giving to υG the topological group structure. We also prove that a C-compact subset A of a topological group G is relatively pseudocompact in the subspace B = A · A −1 · A of G.

Direct sums and products in topological groups and vector spaces

Journal of Mathematical Analysis and Applications, 2016

We call a subset A of an abelian topological group G: (i) absolutely Cauchy summable provided that for every open neighbourhood U of 0 one can find a finite set F ⊆ A such that the subgroup generated by A \ F is contained in U ; (ii) absolutely summable if, for every family {za : a ∈ A} of integer numbers, there exists g ∈ G such that the net a∈F zaa : F ⊆ A is finite converges to g; (iii) topologically independent provided that 0 ∈ A and for every neighbourhood W of 0 there exists a neighbourhood V of 0 such that, for every finite set F ⊆ A and each set {za : a ∈ F } of integers, a∈F zaa ∈ V implies that zaa ∈ W for all a ∈ F. We prove that: (1) an abelian topological group contains a direct product (direct sum) of κ-many non-trivial topological groups if and only if it contains a topologically independent, absolutely (Cauchy) summable subset of cardinality κ; (2) a topological vector space contains R (N) as its subspace if and only if it has an infinite absolutely Cauchy summable set; (3) a topological vector space contains R N as its subspace if and only if it has an R N multiplier convergent series of non-zero elements. We answer a question of Hušek and generalize results by Bessaga-Pelczynski-Rolewicz, Dominguez-Tarieladze and Lipecki.