Free Topological Groups Research Articles

D-spaces and covering properties

We study the D-space property and its generalizations, the notions of an aD-space and a weak aD-space in connection with covering properties. A brief survey on D-spaces is presented in Section 1. Among new results, it is proved that if a linearly ordered space is an aD-space, then it is paracompact. The statement further extends the list of equivalences in [Proc. Amer. Math. Soc. 125 (1997) 1237]. We also establish some sufficient conditions for the free topological group of a Tychonoff space to be a D-space. In particular, the free topological group of a semi-stratifiable space is shown to be a D-space, while it need not be semi-stratifiable. A similar result is established for the free topological group of a space with a point-countable base. Some new interesting open problems on D-spaces and on spaces close to them are formulated. In particular, we discuss several such questions in connection with the sum theorems.

The natural mappings in and k-subspaces of free topological groups on metrizable spaces

Let F( X) be the free topological group on a Tychonoff space X. For all natural number n we denote by F n ( X) the subset of F( X) consisting of all words of reduced length ⩽ n, and by i n the natural mapping from ( X⊕ X −1⊕{ e}) n to F n ( X). We prove that for a metrizable space X if F n ( X) is a k-space for each n, then X is locally compact and either separable or discrete. Therefore, as a corollary, we obtain that for a metrizable space X if F n ( X) is a k-space for all n∈ N , then so is F( X). Furthermore, it is proved that for a metrizable space X the following are equivalent: (i) the mapping i n is a quotient mapping for each n; (ii) a subset U of F( X) is open if i n −1( U∩ F n ( X)) is open in ( X⊕ X −1⊕{ e}) n for each n; (iii) X is locally compact separable or discrete.

Local compactness in free topological groups

We show that the subspace An(X) of the free Abelian topological group A(X) on a Tychonoff space X is locally compact for each n ∈ ω if and only if A2(X) is locally compact if an only if F2(X) is locally compact if and only if X is the topological sum of a compact space and a discrete space. It is also proved that the subspace Fn(X) of the free topological group F(X) is locally compact for each n ∈ ω if and only if F4(X) is locally compact if and only if Fn(X) has pointwise countable type for each n ∈ ω if and only if F4(X) has pointwise countable type if and only if X is either compact or discrete, thus refining a result by Pestov and Yamada. We further show that An(X) has pointwise countable type for each n ∈ ω if and only if A2(X) has pointwise countable type if and only if F2(X) has pointwise countable type if and only if there exists a compact set C of countable character in X such that the complement X \ C is discrete. Finally, we show that F2(X) is locally compact if and only if F3(X) is locally compact, and that F2(X) has pointwise countable type if and only if F3(X) has pointwise countable type.

On the Dimension Modulo Classes of Topological Spaces and Free Topological Groups

Abstract Functions of dimension modulo a (rather wide) class of spaces are considered and the conditions are found, under which the dimension of the product of spaces modulo these classes is equal to zero. Based on these results, the sufficient conditions are established, under which spaces of free topological semigroups (in the sense of Marxen) and spaces of free topological groups (in the sense of Markov and Graev) are zero-dimensional modulo classes of compact spaces.

The free abelian topological group as a subgroup of the free locally convex topological vector space

Article The free abelian topological group as a subgroup of the free locally convex topological vector space was published on March 27, 2003 in the journal Journal of Group Theory (volume 6, issue 3).

Fréchet-Urysohn spaces in free topological groups

Let F(X) and A(X) be respectively the free topological group and the free Abelian topological group on a Tychonoff space X. For every natural number n we denote by F n (X) (A n (X)) the subset of F(X) (A(X)) consisting of all words of reduced length 3 except for n = 4. In addition, however, there is a first countable, and hence Frechet-Urysohn subspace Y of F(X) (A(X)) which is not contained in any F n (X) (A n (X)). We shall show that if such a space Y is first countable, then it has a special form in F(X) (A(X)). On the other hand, we give an example showing that if the space Y is Frechet-Urysohn, then it need not have the form.

Provisional solution to a Comfort–van Mill problem

We prove under the assumption of Martin's Axiom that an abstract Abelian group G of non-measurable cardinality is the intersection of countably compact subgroups of its Bohr compactification bG. This result is used to show that weakly free countably compact topological groups do not exist, thus answering a question posed by Comfort and van Mill in 1988. In fact, we show under MA that a free ( P, CC )-group over a space X exists iff X is empty, where P and CC are the classes of pseudocompact and countably compact topological groups, respectively. On the other hand, we prove the existence of a weakly free ( P, CC )-group over an arbitrary space X and show that our construction of such a group is functorial. Similar results remain valid in the Abelian case.

Weakly complete free topological groups

A topological group G is sequentially complete if it is sequentially closed in any other topological group. We show that for a Tychonoff space X, the free topological group F( X) is sequentially complete iff the free Abelian topological group A( X) is sequentially complete iff X is sequentially closed in βX. Furthermore, the free precompact Abelian group F(X, PA ) is sequentially complete iff the space X is sequentially closed in βX. We consider also other forms of weak completeness, namely ω-completeness and k-completeness, introduced analogously by means of the ω-closure and the k-closure. We prove that the groups A( X) and F( X) are ω-complete ( k-complete) iff X is ω-closed ( k-closed) in the Dieudonné completion μX of X.

Free topological groups of weak [formula omitted]-spaces

It is known that the free topological group over the Tychonoff space X, denoted F(X), is a P-space if and only if X is a P-space. This article is concerned with the question of whether one can characterize when F(X) is a weak P-space, that is, a space where all countable subsets are closed. Our main result is that F(X) is a weak P-space if and only if X is a weak P-space and every countable subset is C-embedded.

Some properties of o-bounded and strictly o-bounded groups

<p>We continue the study of (strictly) o-bounded topological groups initiated by the first listed author and solve two problems posed earlier. It is shown here that the product of a Comfort-like topological group by a (strictly) o-bounded group is (strictly) o-bounded. Some non-trivial examples of strictly o-bounded free topological groups are given. We also show that o-boundedness is not productive, and strict o-boundedness cannot be characterized by means of second countable continuous homomorphic images.</p><p> </p>

τ-locally invariant groups

A class of τ -locally invariant topological groups is introduced; this class is a new one for τ a limit cardinal, and it coincides with Arhangel'skiı̆'s class of λ -balanced groups for τ = λ + . It is proved that a topological group is τ -locally invariant if and only if this group is topologically isomorphic to a subgroup of a direct product of groups of topological character less than τ . Topological spaces whose free topological groups are τ -locally invariant are described.

Completeness of hyperspaces on topological groups

We prove that if the bilateral uniformity B of a topological group of pointwise countable type is complete, then the Hausdorff–Bourbaki uniformity of B is complete. It follows that for every Čech complete topological group, the Hausdorff–Bourbaki uniformity of B is complete. Finally, we prove that if X is a compact topological space, then the Hausdorff–Bourbaki uniformity of the bilateral uniformity of the free Abelian topological group over X, is complete.

Essential subgroups of topological groups

We study the class Wof Hausdorff topological groups Gfor which the following two cardinal invariants coincide ES(G)=min{|H|:H≤ Gdense and essential} TD(G)=min{|H|:H≤ Gtotally dense} We prove that W contains the following classes:locally compact abelian groups, compact connected groups, countably compact totally discon¬nected abelian groups, topologically simple groups, locally compact Abelian groups when endowed with their Bohr topology, totally minimal abelian groups and free Abelian topological groups. For all these classes we are also able to giv ean explicit computation of the common value of ESand TD.

Free topological groups of spaces and their subspaces

We prove that if X is a Tychonoff topological space, Y a subspace of X, and every bounded continuous pseudometric on Y can be extended to a continuous pseudometric on X, then the free topological group F M ( Y) coincides with the topological subgroup of F M ( X) generated by Y. For this purpose, a new description for the topology of a free topological group in terms of continuous pseudometrics and group seminorms is given. It follows from what has been shown by Uspenskiı̆ that this result implies the Weil completeness of F M ( X) for any Dieudonné complete X. It is also proved that if dim X=0 , then ind F M(X)=0 .

Free topological groups on metrizable spaces and inductive limits

We prove that for a metrizable space X the following are equivalent: (i) the free Abelian topological group A(X) is the inductive limit of the sequence {An(X):n∈N}, where An(X) is formed by all words of reduced length ≤n; (ii) X is locally compact and the set of all non-isolated points of X is separable. In the non-Abelian case, for a metrizable X the following are equivalent: (i) the free topological group F(X) is the inductive limit of the sequence {Fn(X):n∈N}; (ii) X is either locally compact separable or discrete.

Some topological groups with and some without suitable sets

If a discrete subset S of a topological group G with identity 1 generates a dense subgroup of G and S∪{1} is closed in G, then S is called a suitable set for G. We construct in ZFC a Lindelöf topological group L such that t(L)·ψ(L)≤ℵ0 and L does not have a suitable set. We also give a ZFC example of a countably compact topological group H with no suitable set; in addition, the closure of every countable subset of H is compact. It is proved that a non-pseudocompact topological group with a dense strictly σ-discrete subset has a closed suitable set. This implies, in particular, that a free (Abelian) topological group on a metrizable space has a closed suitable set.

Suitable sets for topological groups

A subset S of a topological group G is said to be a suitable set if (a) it has the discrete topology, (b) it is a closed subset of G{1}, and (c) the subgroup generated by S is dense in G. K.H. Hoffmann and S.A. Morris proved that every locally compact group has a suitable set. In this paper it is proved that every metrizable topological group and every countable Hausdorff topological group has a suitable set. Examples of Hausdorff topological groups without suitable sets are produced. The free abelian topological group on the Stone-Čech compactification of any uncountable discrete space is one such example. Under the assumption of the Continuum Hypothesis or Martin's Axiom it is shown that examples exist of separable Hausdorff topological groups with no suitable set. It is not known if such examples exist in ZFC alone. An example is produced here of a compact connected abelian group with a one-element suitable set which has a dense σ-compact connected subgroup with no suitable set.

Quotients of zero-dimensional precompact abelian groups

Shakhmatov (1990) proved that precompact groups can be presented as quotients of zerodimensional precompact ones with preservation of the weight. We obtain here a new proof of this result in the abelian case making no use of free topological groups. We study the question when various properties (total minimality, being torsion-free, etc.) can be preserved under this process of lifting.

A relatively free topological group that is not varietal free

Answering a 1982 question of Sidney A. Morris, we construct a topological group G and a subspace X such that (i) G is algebraically free over X, (ii) G is relatively free over X, that is, every continuous mapping from X to G extends to a unique continuous

The Free Abelian Topological Group and the Free Locally Convex Space on the Unit Interval

We give a complete description of the topological spaces X such that the free abelian topological group A(X) embeds into the free abelian topological group A(I) on the closed unit interval. In particular, the free abelian topological group A(X) on any finite-dimensional compact metrizable space X embeds into A(I). To obtain our description, we study similar embeddings of the free locally convex spaces and continuous surjections between the spaces of continuous functions with the pointwise topology. Proofs are based on the classical Kolmogorov's Superposition Theorem.