Free Topological Groups Research Articles

THE SHAPE OF COMPACT COVERS

Abstract For a space X let $\mathcal {K}(X)$ be the set of compact subsets of X ordered by inclusion. A map $\phi :\mathcal {K}(X) \to \mathcal {K}(Y)$ is a relative Tukey quotient if it carries compact covers to compact covers. When there is such a Tukey quotient write $(X,\mathcal {K}(X)) \ge _T (Y,\mathcal {K}(Y))$ , and write $(X,\mathcal {K}(X)) =_T (Y,\mathcal {K}(Y))$ if $(X,\mathcal {K}(X)) \ge _T (Y,\mathcal {K}(Y))$ and vice versa. We investigate the initial structure of pairs $(X,\mathcal {K}(X))$ under the relative Tukey order, focussing on the case of separable metrizable spaces. Connections are made to Menger spaces. Applications are given demonstrating the diversity of free topological groups, and related free objects, over separable metrizable spaces. It is shown a topological group G has the countable chain condition if it is either $\sigma $ -pseudocompact or for some separable metrizable M, we have $\mathcal {K}(M) \ge _T (G,\mathcal {K}(G))$ .

A survey of generalized metrizable properties in topological groups and weakly topological groups

The theory of generalized metrizable spaces is an important topic of general topology. This paper is a survey of research methods and achievements on generalized metrizable properties in topological groups and weakly topological groups. We mainly study this kind of properties in topological groups, semitopological groups, paratopological groups, quasitopological groups and free topological groups, and focus on the influence of separation properties, conditions for weakly topological groups to become topological groups, cardinal invariants, weak first-countability, three-space properties and remainders in compactifications on topological groups and related structures. Finally, some unsolved problems in this field are listed for researchers.

Real-valued measurable cardinals and sequentially continuous homomorphisms

A. V. Arkhangel'skiĭ asked in 1981 if the variety V of topological groups generated by free topological groups on metrizable spaces coincides with the class of all topological groups. We show that if there exists a real-valued measurable cardinal then the variety V is a proper subclass of the class of all topological groups. A topological group G is called g-sequential if for any topological group H any sequentially continuous homomorphism G→H is continuous. We introduce the concept of a g-sequential cardinal and prove that a locally compact group is g-sequential if and only if its local weight is not a g-sequential cardinal. The product of a family of non-trivial g-sequential topological groups is g-sequential if and only if the cardinal of this family is not g-sequential. Suppose G is either the unitary group of a Hilbert space or the group of all self-homeomorphisms of a Tikhonov cube. Then G is g-sequential if and only if its weight is not a g-sequential cardinal. Every compact group of Ulam-measurable cardinality admits a strictly finer countably compact group topology.

Topological groups with strong disconnectedness properties

Topological groups whose underlying spaces are basically disconnected, F-, or F′-spaces but not P-spaces are considered. It is proved, in particular, that the existence of a Lindelöf basically disconnected topological group which is not a P-space is equivalent to the existence of a Boolean basically disconnected Lindelöf group of countable pseudocharacter, that free and free Abelian topological groups of zero-dimensional non-P-spaces are never F′-spaces, and that the existence of a free Boolean F′-group which is not a P-space is equivalent to that of selective ultrafilters on ω.

Is the Free Locally Convex Space L(X) Nuclear?

Given a class $${{\mathcal {P}}}$$ of Banach spaces, a locally convex space (LCS) E is called multi- $${{\mathcal {P}}}$$ if E can be isomorphically embedded into a product of spaces that belong to $${{\mathcal {P}}}$$ . We investigate the question whether the free locally convex space L(X) is strongly nuclear, nuclear, Schwartz, multi-Hilbert or multi-reflexive. If X is a Tychonoff space containing an infinite compact subset then, as it follows from the results of Außenhofer (Topol Appl 134:90–102, 2007), L(X) is not nuclear. We prove that for such X the free LCS L(X) has the stronger property of not being multi-Hilbert. We deduce that if X is a k-space, then the following properties are equivalent: (1) L(X) is strongly nuclear; (2) L(X) is nuclear; (3) L(X) is multi-Hilbert; (4) X is countable and discrete. On the other hand, we show that L(X) is strongly nuclear for every projectively countable P-space (in particular, for every Lindelöf P-space) X. We observe that every Schwartz LCS is multi-reflexive. It is known that if X is a $$k_\omega $$ -space, then L(X) is a Schwartz LCS (Außenhofer et al. in Stud Math 181(3):199–210, 2007), hence L(X) is multi-reflexive. We show that for every first-countable paracompact (in particular, for every metrizable) space X the converse is true, so L(X) is multi-reflexive if and only if X is a $$k_\omega $$ -space, equivalently if X is a locally compact and $$\sigma $$ -compact space. Similarly, we show that for any first-countable paracompact space X the free abelian topological group A(X) is a Schwartz group if and only if X is a locally compact space such that the set $$X^{(1)}$$ of all non-isolated points of X is $$\sigma $$ -compact.

Products of topological groups and ω-stability

We present conditions on factors guaranteeing that the product of two spaces (topological groups) is (fairly) pseudo-ℵ1-compact or ω-stable. It is shown, for example, that the product X×Y of a regular pseudo-ℵ1-compact P-space X with a weakly Lindelöf space Y is fairly pseudo-ℵ1-compact. Similarly, the product G×Y of a pseudo-ℵ1-compact P-group G and a fairly pseudo-ℵ1-compact space Y is fairly pseudo-ℵ1-compact.Also, we prove that for each infinite cardinal τ, a Tychonoff space X is τ-stable if and only if the free topological group F(X) (equivalently, the free Abelian topological group A(X)) on X is τ-steady. This enables us to establish τ-stability of several classes of Tychonoff spaces. Additionally, we resolve Open Problem 5.6.2 from [2] by proving that the product G×H of a Lindelöf Σ-group G and a Lindelöf P-group H is τ-stable provided τ=ℵn for some n∈ω or τω=τ.

A note on radical and radical free topological groups

A note on radical and radical free topological groups

Topological groups of Lipschitz functions and Graev metrics

We study the properties of the free abelian topological group Ad(X) on a metric space (X,d) endowed with the topology generated by the Graev extension dˆ of a given metric d on X. We find that the group of Lipschitz functions Lip0(X,T) is the group of continuous characters of Ad(X). From this fact we derive some interesting properties of the metric groups Ad(X) and Lip0(X,T).

$P$-bases and topological groups

A topological space $X$ is defined to have a neighborhood $P$-base at any $x\in X$ from some partially ordered set (poset) $P$ if there exists a neighborhood base $(U_p[x])_{p\in P}$ at $x$ such that $U_p[x]\subseteq U_{p’}[x]$ for all $p\geq p’$ in $P$. We prove that a compact space is countable, hence metrizable, if it has countable scattered height and a $\mathcal {K}(M)$-base for some separable metric space $M$. Banakh [Dissertationes Math. 538 (2019), p. 141] gives a positive answer to Problem 8.6.8. Let $A(X)$ be the free Abelian topological group on $X$. It is shown that if $Y$ is a retract of $X$ such that the free Abelian topological group $A(Y)$ has a $P$-base and $A(X/Y)$ has a $Q$-base, then $A(X)$ has a $P\times Q$-base. Also if $Y$ is a closed subspace of $X$ and $A(X)$ has a $P$-base, then $A(X/Y)$ has a $P$-base. It is shown that any Fréchet-Urysohn topological group with a $\mathcal {K}(M)$-base for some separable metric space $M$ is first-countable, hence metrizable. And if $P$ is a poset with calibre $(\omega _1, \omega )$ and $G$ is a topological group with a $P$-base, then any precompact subset in G is metrizable, hence $G$ is strictly angelic. Applications in function spaces $C_p(X)$ and $C_k(X)$ are discussed. We also give an example of a topological Boolean group of character $\leq \mathfrak {d}$ such that the precompact subsets are metrizable but $G$ doesn’t have an $\omega ^\omega$-base if $\omega _1<\mathfrak {d}$. Gabriyelyan, Kakol, and Liederman [Fund. Math. 229 (2015), pp. 129–158] give a consistent negative answer to Problem 6.5.

Tukey order and diversity of free Abelian topological groups

For a Tychonoff space X the free Abelian topological group over X , denoted A ( X ) , is the free Abelian group on the set X with the coarsest topology so that for any continuous map of X into an Abelian topological group its canonical extension to a homomorphism on A ( X ) is continuous. We show there is a family A of maximal size, 2 c , consisting of separable metrizable spaces , such that if M and N are distinct members of A then A ( M ) and A ( N ) are not topologically isomorphic (moreover, A ( M ) neither embeds topologically in A ( N ) nor is an open image of A ( N ) ). We show there is a chain C = { M α : α < c + } , of maximal size, of separable metrizable spaces such that if β < α then A ( M β ) embeds as a closed subgroup of A ( M α ) but no subspace of A ( M β ) is homeomorphic to A ( M α ) . We show that the character (minimal size of a local base at 0) of A ( M ) is d (minimal size of a cofinal set in N N ) for every non-discrete, analytic M , but consistently there is a co-analytic M such that the character of A ( M ) is strictly above d . The main tool used for these results is the Tukey order on the neighborhood filter at 0 in an A ( X ) , and a connection with the family of compact subsets of an auxiliary space.

Topological groups with invariant linear spans

Given a topological group G that can be embedded as a topological subgroup into some topological vector space (over the field of reals) we say that G has invariant linear span if all linear spans of G under arbitrary embeddings into topological vector spaces are isomorphic as topological vector spaces. For an arbitrary set A let \({{\mathbb {Z}}}^{(A)}\) be the direct sum of |A|-many copies of the discrete group of integers endowed with the Tychonoff product topology. We show that the topological group \({{\mathbb {Z}}}^{(A)}\) has invariant linear span. This answers a question from a paper of Dikranjan et al. (J Math Anal Appl 437:1257–1282, 2016) in positive. We prove that given a non-discrete sequential space X, the free abelian topological group A(X) over X is an example of a topological group that embeds into a topological vector space but does not have invariant linear span.

Fréchet-Urysohn subspaces of free topological groups, II

Let F(X) be the free topological group on a Tychonoff space X. For all natural numbers n we denote by Fn(X) the subset of F(X) consisting of all words of reduced length ≤n. In [9], the author found equivalent conditions on a metrizable space X for F3(X) to be Fréchet-Urysohn, and for Fn(X) to be Fréchet-Urysohn for n≥5. However, no equivalent condition on X for n=4 was found. Recently, we proved in [11] that if a metrizable space X is locally compact separable and the set of all non-isolated points of X is compact, then F4(X) is Fréchet-Urysohn. In this paper we continue these investigations and show that ‘separability’ of X can be omitted. The result is an affirmative answer to Question 2 in [11] as well as Question 4.2 in [4].

Long colimits of topological groups II: Free groups and vector spaces

Topological properties of the free topological group and the free abelian topological group on a space have been thoroughly studied since the 1940s. In this paper, we study the free topological R-vector space V(X) on X. We show that V(X) is a quotient of the free abelian topological group on [−1,1]×X, and use this to prove topological vector space analogues of existing results for free topological groups on pseudocompact spaces. As an application, we show that certain families of subspaces of V(X) satisfy the so-called algebraic colimit property defined in the authors' previous work.

Metrizable quotients of free topological groups

We consider the following problem: For which Tychonoff topological spaces X do the free topological group F(X) and the free abelian topological group A(X) admit a nontrivial (i.e. not finitely generated) metrizable quotient group? First, we give a complete solution of this problem for the key class of compact spaces X. Then, relying on this result, we resolve the problem for several more general important classes of spaces X, including the class of $$\sigma $$ -compact spaces, the class of pseudocompact spaces, the class of $$\omega $$ -bounded spaces, the class of Cech-complete spaces and the class of K-analytic spaces. Also we describe all absolutely analytic metric spaces X (in particular, completely metrizable spaces) such that the free topological group F(X) and the free abelian topological group A(X) admit a nontrivial metrizable quotient group. Our results are based on an extensive use of topological properties of non-scattered spaces.

The Dieudonné τ-complete spaces and free topological groups of uniform spaces

For a Tychonoff space X, the Dieudonné τ-completion of X, denoted by μτX, is investigated. The space μτX is defined as the completion of X with respect to the uniformity uXτ, where uXτ is generated by all continuous mappings of X to metric spaces of weight ≤τ. It is proved that a dense subspace Y of X is Pτ-(equivalently, Pτ⁎-)embedded in X iff uXτ|Y=uYτ iff Y⊂X⊂μτX. In this case, the free topological group F(uYτY) of the uniform space uYτY is the topological subgroup of the group F(uXτX) generated by Y iff Y is Pτ-embedded in X. This result implies the Nummela-Pestov's Theorem: A dense subspace Y is P-embedded in a Tychonoff space X iff the free topological group F(Y) is topologically isomorphic to the subgroup of the group F(X) generated by Y. It is proved that the Weil completion of the free topological group coincides with its Raĭkov completion. It is also shown that the free topological groups in the sense of Nakayama and Nummela coincide.

Countable tightness and $${\mathfrak {G}}$$-bases on free topological groups

Given a Tychonoff space X, let F(X) and A(X) be respectively the free topological group and the free Abelian topological group over X in the sense of Markov. In this paper, we consider two topological properties of F(X) or A(X), namely the countable tightness and $$\mathfrak G$$-base. We provide some characterizations of the countable tightness and $$\mathfrak G$$-base of F(X) and A(X) for various special classes of spaces X. Furthermore, we also study the countable tightness and $$\mathfrak G$$-base of some $$F_{n}(X)$$ of F(X).

On the isomorphic classification of the free topological groups of the non-Tychonoff spaces

In the paper we proposed the method for reducing the problem of isomorphic classi­fication of free topological groups of completely regular spaces to the problem of isomorphic classification of free topological groups of completely Tychonoff spaces. Cite as: N. M. Pyrch, “On the isomorphic classification of the free topological groups of the non-Tychonoff spaces,” Prykl. Probl. Mekh. Mat. , Issue 17, 27–37 (2019) (in Ukrainian), https://doi.org/10.15407/apmm2019.17.27-37

Separable Quotients of Free Topological Groups

AbstractWe study the following problem: For which Tychonoff spaces $X$ do the free topological group $F(X)$ and the free abelian topological group $A(X)$ admit a quotient homomorphism onto a separable and nontrivial (i.e., not finitely generated) group? The existence of the required quotient homomorphisms is established for several important classes of spaces $X$, which include the class of pseudocompact spaces, the class of locally compact spaces, the class of $\unicode[STIX]{x1D70E}$-compact spaces, the class of connected locally connected spaces, and some others.We also show that there exists an infinite separable precompact topological abelian group $G$ such that every quotient of $G$ is either the one-point group or contains a dense non-separable subgroup and, hence, does not have a countable network.

An affirmative answer to Yamada's Conjecture

In this paper, we give an affirmative answer to Yamada's Conjecture on free topological groups, which was posed in K. Yamada (2002) [11].

A REMARK ON THE SEPARABLE QUOTIENT PROBLEM FOR TOPOLOGICAL GROUPS

The Banach–Mazur separable quotient problem asks whether every infinite-dimensional Banach space $B$ has a quotient space that is an infinite-dimensional separable Banach space. The question has remained open for over 80 years, although an affirmative answer is known in special cases such as when $B$ is reflexive or even a dual of a Banach space. Very recently, it has been shown to be true for dual-like spaces. An analogous problem for topological groups is: Does every infinite-dimensional (in the topological sense) connected (Hausdorff) topological group $G$ have a quotient topological group that is infinite dimensional and metrisable? While this is known to be true if $G$ is the underlying topological group of an infinite-dimensional Banach space, it is shown here to be false even if $G$ is the underlying topological group of an infinite-dimensional locally convex space. Indeed, it is shown that the free topological vector space on any countably infinite $k_{\unicode[STIX]{x1D714}}$-space is an infinite-dimensional toplogical vector space which does not have any quotient topological group that is infinite dimensional and metrisable. By contrast, the Graev free abelian topological group and the Graev free topological group on any infinite connected Tychonoff space, both of which are connected topological groups, are shown here to have the tubby torus $\mathbb{T}^{\unicode[STIX]{x1D714}}$, which is an infinite-dimensional metrisable group, as a quotient group.