Free Topological Groups Research Articles

Topologies on groups determined by sets of convergent sequences

A Hausdorff topological group (G,τ) is called an s-group and τ is called an s-topology if there is a set S of sequences in G such that τ is the finest Hausdorff group topology on G in which every sequence of S converges to the unit. The class S of all s-groups contains all sequential Hausdorff groups and it is finitely multiplicative. A quotient group of an s-group is an s-group. For a non-discrete topological group (G,τ) the following three assertions are equivalent: (1) (G,τ) is an s-group, (2) (G,τ) is a quotient group of a Graev free topological group over a metrizable space, (3) (G,τ) is a quotient group of a Graev free topological group over a sequential Tychonoff space. The Abelian version of this characterization of s-groups holds as well.

A note on free paratopological groups

In this paper, we mainly discuss some generalized metric properties and the character of the free paratopological groups, and extend several results valid for free topological groups to free paratopological groups.

Coarse structures on groups

We introduce the group-compact coarse structure on a Hausdorff topological group in the context of coarse structures on an abstract group which are compatible with the group operations. We develop asymptotic dimension theory for the group-compact coarse structure generalizing several familiar results for discrete groups. We show that the asymptotic dimension in our sense of the free topological group on a non-empty topological space that is homeomorphic to a closed subspace of a Cartesian product of metrizable spaces is 1.

On the topology of free paratopological groups

The result often known as Joiner's lemma is fundamental in understanding the topology of the free topological group $F(X)$ on a Tychonoff space$X$. In this paper, an analogue of Joiner's lemma for the free paratopological group $\FP(X)$ on a $T_1$ space $X$ is proved. Using this, it is shown that the following conditions are equivalent for a space $X$: (1) $X$ is $T_1$; (2) $\FP(X)$ is $T_1$; (3) the subspace $X$ of $\FP(X)$ is closed; (4) the subspace $X^{-1}$ of $\FP(X)$ is discrete; (5) the subspace $X^{-1}$ is $T_1$; (6) the subspace $X^{-1}$ is closed; and (7) the subspace $\FP_n(X)$ is closed for all $n \in \N$, where $\FP_n(X)$ denotes the subspace of $\FP(X)$ consisting of all words of length at most $n$.

Free abelian topological groups and collapsing maps

We show that for every pair (X,Y) of ANR compacta, Y⊂X, the free abelian topological group applied to a collapsing map q:X→X/Y produces a locally trivial bundle A(q):A(X)→A(X/Y) with the fiber A(Y). As a result we obtain a short proof of the classical Dold–Thom theorem which states that H˜n(X)=πn(A(X)) for all complexes X.

The topological fundamental group and free topological groups

The topological fundamental group π 1 top is a homotopy invariant finer than the usual fundamental group. It assigns to each space a quasitopological group and is discrete on spaces which admit universal covers. For an arbitrary space X, we compute the topological fundamental group of the suspension space Σ ( X + ) and find that π 1 top ( Σ ( X + ) ) either fails to be a topological group or is the free topological group on the path component space of X. Using this computation, we provide an abundance of counterexamples to the assertion that all topological fundamental groups are topological groups. A relation to free topological groups allows us to reduce the problem of characterizing Hausdorff spaces X for which π 1 top ( Σ ( X + ) ) is a Hausdorff topological group to some well-known classification problems in topology.

O-Boundedness of free topological groups

Assuming the absence of Q-points (which is consistent with ZFC) we prove that the free topological group F ( X ) over a Tychonov space X is o-bounded if and only if every continuous metrizable image T of X satisfies the selection principle ⋃ fin ( O , Ω ) (the latter means that for every sequence 〈 u n 〉 n ∈ ω of open covers of T there exists a sequence 〈 v n 〉 n ∈ ω such that v n ∈ [ u n ] < ω and for every F ∈ [ X ] < ω there exists n ∈ ω with F ⊂ ⋃ v n ). This characterization gives a consistent answer to a problem posed by C. Hernándes, D. Robbie, and M. Tkachenko in 2000.

Metrization of free groups on ultrametric spaces

We consider ultrametrizations of free topological groups of ultrametric spaces. A construction is defined that determines a functor in the category UMET 1 of ultrametric spaces of diameter ⩽1 and nonexpanding maps. This functor is the functorial part of a monad in UMET 1 and we provide a characterization of the category of its algebras.

On unitary representability of topological groups

We prove that the additive group (E*, τ k (E)) of an $${\mathcal {L}_\infty}$$ -Banach space E, with the topology τ k (E) of uniform convergence on compact subsets of E, is topologically isomorphic to a subgroup of the unitary group of some Hilbert space (is unitarily representable). This is the same as proving that the topological group (E*, τ k (E)) is uniformly homeomorphic to a subset of $${\ell_2^\kappa}$$ for some κ. As an immediate consequence, preduals of commutative von Neumann algebras or duals of commutative C*-algebras are unitarily representable in the topology of uniform convergence on compact subsets. The unitary representability of free locally convex spaces (and thus of free Abelian topological groups) on compact spaces, follows as well. The above facts cannot be extended to noncommutative von Neumann algebras or general Schwartz spaces.

VARIETIES OF ABELIAN TOPOLOGICAL GROUPS AND SCATTERED SPACES

AbstractThe variety of topological groups generated by the class of all abelian kω-groups has been shown to equal the variety of topological groups generated by the free abelian topological group on [0, 1]. In this paper it is proved that the free abelian topological group on a compact Hausdorff space X generates the same variety if and only if X is not scattered.

Unitary representability of free abelian topological groups

For every Tikhonov space X the free abelian topological group A(X) and the free locally convex vector space L(X) admit a topologically faithful unitary representation. For compact spaces X this is due to Jorge Galindo.

Fréchet–Urysohn fans in free topological groups

In this paper we answer the question of T. Banakh and M. Zarichnyi constructing a copy of the Fréchet–Urysohn fan S ω in a topological group G admitting a functorial embedding [ 0 , 1 ] ⊂ G . The latter means that each autohomeomorphism of [ 0 , 1 ] extends to a continuous homomorphism of G . This implies that many natural free topological group constructions (e.g. the constructions of the Markov free topological group, free abelian topological group, free totally bounded group, free compact group) applied to a Tychonov space X containing a topological copy of the space Q of rationals give topological groups containing S ω .

Graev metric groups and Polishable subgroups

We prove a conjecture of Hjorth: There is an uncountable Polish group all of whose abelian subgroups are discrete. We first construct directly a witness to Hjorth's conjecture. Then we consider an existing example in the literature. The example is the metric completion of a free topological group constructed by Graev. We give a definition slightly more general than Graev's and prove some properties of the Graev metrics which seem to be unknown previously. We also consider the problem of finding Polishable subgroups of the Graev metric groups with arbitrarily high Borel rank. In doing this we prove some general theorems on extensions of Polish groups with this property.

On the pull-back of metabelian topological groups

Let Q be a topological group and N a trivial Q-module . A topological extension of Q by N is a short exact sequence 0 → N i → G π → Q → 0 ,where i is a topological embedding onto a closed subgroup and π an open continuous onto homomorphism. The continuous map u : Q → G is a section if πu = IdQ. All topological spaces are assumed to be Tychanov (completely regular , Hausdorff). The Free topological group is in Markov sense [9]. If X is completely regular ,then the (Markov) free topological group on X is the group F (X) equipped with the finest group topology inducing the given topology on X as a subspace. Such a topology always exists [9], and has the universal property of the following kind:every continuous mapping f from X to an arbitrary topological group G lifts to a unique continuous homomorphism

On Schwartz groups

We introduce a notion of a Schwartz group, which turns out to be coherent with the well known concept of a Schwartz topological vector space. We establish several basic properties of Schwartz groups and show that free topological Abelian groups, as well a

Free nuclear groups

We define free groups in certain classes of abelian topological groups. Of special interest is the class of nuclear groups. For a completely regular Hausdorff space X we show that the free nuclear group A N ( X ) (i.e., the free abelian group in the class of all nuclear groups) exists. We give two different descriptions of the topology on A N ( X ) . Afterwards we study the question under which conditions the free abelian topological group A ( X ) is topologically isomorphic to the free nuclear group A N ( X ) and show that if X is a product of metrizable spaces then this is true if and only if X is discrete.

Dieudonné completion and [formula omitted]-group actions

If G ( X ) denotes either the free topological group or the free Abelian topological group over a topological space X, we prove that ∏ i = 1 n G ( X i ) is a hemibounded b f -group whenever each X i is a pseudocompact space (which provides a new way to generate this kind of topological groups), and we show that the equality μ ( X × ∏ i = 1 n G ( X i ) ) = μ X × ∏ i = 1 n G ( β X i ) holds whenever X is a hemibounded b f -space (where μY stands for the Dieudonné completion of Y). By means of the Dieudonné completion we prove that every pseudocompact space X is G-Tychonoff whenever G is a b f -group and that the maximal G-compactification of X coincides with βX. We apply this result to obtain a partial version for G-spaces of Glicksberg's theorem on pseudocompactness and we analyze when the maximal G-compactification of a G-space X coincides with the Stone–Čech compactification of X in the case when G is a metrizable group.

The Topology of Free Topological Groups

The known explicit descriptions of the topology of free and free Abelian topological groups are collected. Examples of the application of such descriptions by various authors to study the properties of topological groups and some related topological-algebraic objects are given. Various approaches to describing the topology of free topological groups are presented and a general method for topologizing free groups is suggested.The fundamental properties of free topological groups and their structure are considered.The most important results on free topological groups obtained by various authors through the years are mentioned.

The character of free topological groups I

A systematic analysis is made of the character of the free and free abelian topological groups on uniform spaces and on topological spaces. In the case of the free abelian topological group on a uniform space, expressions are given for the character in terms of simple cardinal invariants of the family of uniformly continuous pseudometrics of the given uniform space and of the uniformity itself. From these results, others follow on the basis of various topological assumptions. Amongst these: (i) if X is a compact Hausdorff space, then the character of the free abelian topological group on X lies between w(X) and w(X)ℵ0, where w(X) denotes the weight of X; (ii) if the Tychonoff space X is not a P-space, then the character of the free abelian topological group is bounded below by the “small cardinal” d; and (iii) if X is an infinite compact metrizable space, then the character is precisely d. In the non-abelian case, we show that the character of the free abelian topological group is always less than or equal to that of the corresponding free topological group, but the inequality is in general strict. It is also shown that the characters of the free abelian and the free topological groups are equal whenever the given uniform space is w-narrow. A sequel to this paper analyses more closely the cases of the free and free abelian topological groups on compact Hausdorff spaces and metrizable spaces.

The character of free topological groups II

A systematic analysis is made of the character of the free and free abelian topological groups on metrizable spaces and compact spaces, and on certain other closely related spaces. In the first case, it is shown that the characters of the free and the free abelian topological groups on X are both equal to the “small cardinal” d if X is compact and metrizable, but also, more generally, if X is a non-discrete k!-space all of whose compact subsets are metrizable, or if X is a non-discrete Polish space. An example is given of a zero-dimensional separable metric space for which both characters are equal to the cardinal of the continuum. In the case of a compact space X, an explicit formula is derived for the character of the free topological group on X involving no cardinal invariant of X other than its weight; in particular the character is fully determined by the weight in the compact case. This paper is a sequel to a paper by the same authors in which the characters of the free groups were analysed under less restrictive topological assumptions.