Strong collective normality and countable compactness in free topological groups (original) (raw)
Local compactness in free topological groups
Bulletin of the Australian Mathematical Society, 2003
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 ...
The character of free topological groups II
Applied General Topology, 2005
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.
The character of free topological groups I
Applied General Topology, 2005
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) ℵ 0 , 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 ω-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.
Free Topological Groups Over Ωµ-Metrizable Spaces
2015
Abstract. Let ωµ be an uncountable regular cardinal. For a Tychonoff space X, we let A(X) and F (X) be the free Abelian topological group and the free topological group over X, respectively. In this paper, we establish the next equivalences. Theorem. Let X be a space. The following are equivalent. 1. (X,UX) is an ωµ-metrizable uniform space, where UX is the universal uniformity on X. 2. A(X) is topologically orderable and χ(A(X)) = ωµ. 3. The derived set Xd is ωµ-compact and X is ωµ-metrizable. Theorem. Let X be a non-discrete space. Then, the following are equiva-lent. 1. X is ωµ-compact and ωµ-metrizable. 2. (X,UX) is ωµ-metrizable and X is ωµ-compact. 3. F (X) is topologically orderable and χ(F (X)) = ωµ. We also prove that an ωµ-metrizable uniform space (X,U) is a retract of its uniform free Abelian group A(X,U) and of its uniform free group F (X,U). 1.
Free topological groups over metrizable spaces
Topology and its Applications, 1989
Let X be a metrizable space and F(X) and A(X) be the free topological group over X and the free Abelian topological group over X respectively. We establish the following criteria:
Two weak forms of countability axioms in free topological groups
Topology and its Applications
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. For every n ∈ N, let Fn(X) (resp. An(X)) denote the subspace of F (X) (resp. A(X)) that consists of words of reduced length at most n with respect to the free basis X. In this paper, we discuss two weak forms of countability axioms in F (X) or A(X), namely the csf-countability and snf-countability. We provide some characterizations of the csf-countability and snf-countability of F (X) and A(X) for various classes of spaces X. In addition, we also study the csf-countability and snf-countability of Fn(X) or An(X), for n = 2, 3, 4. Some results of Arhangel'skiǐ in [1] and Yamada in [22] are generalized. An affirmative answer to an open question posed by Li et al. in [11] is provided.
A remark on free topological groups with no small subgroups
Journal of the Australian Mathematical Society, 1974
For a completely regular space X let G(X) be the Graev free topological group on X. While proving G(X) exists for completely regular spaces X, Graev showed that every pseudo-metric on X can be extended to a two-sided invariant pseudo-metric on the abstract group G(X). The free group topology on G(X) is usually strictly finer than this pseudo-metric topology. In particular this is the case when X is not totally disconnected (see Morris and Thompson [7]). It is of interest to know when G(X) has no small subgroups (see Morris [5]). Morris and Thompson [6] showed that this is the case if and only if X admits a continuous metric. The proof relied on properties of the free group topology and it is natural to ask if G(X) with its pseudo-metric topology has no small subgroups when and only when X admits a continuous metric. We show that this is the case. Topological properties of G(X) associated with the pseudo-metric topology have recently been studied by Joiner [3] and Abels [1].
Normality on Topological Groups
It is a well known fact that every topological group which satisfies a mild separation axiom like being T0, is automatically Hausdorff and completely regular, thus, a Tychonoff space. Further separation axioms do not hold in general. For instance, the topological product of uncountable many copies of the discrete group of integer numbers, say Z R is not normal. Clearly it is a topological Abelian Hausdorff group, with the operation defined pointwise and the product topology τ. With this example in mind, one can ask, are there " many non-normal " groups? Markov asked in 1945 wether every uncountable abstract group admits a non-normal group topology. Van Douwen in 1990 asked if every Abelian group endowed with the weak topology corresponding to the family of all its homomorphisms in the unit circle of the complex plane should be normal. Here we prove that the above group Z R endowed with its Bohr topology τ b is non-normal either, and obtain that all group topologies on Z R which lie between τ b and the original one τ are also non-normal. In fact, every compatible topol-ogy for this group lacks normality and we raise the general question about the " normality behaviour " of compatible group topologies.
Free products of topological groups
Bull. Austral. Math. Soc., 1971
In this note the notion of a free topological product | | G of a set {G } of topological groups is introduced. It is shown that it always exists, is unique and is algebraically isomorphic to the usual free product of the underlying groups. Further if each G is Hausdorff, then ] | G is Hausdorff and each G is a closed subgroup. Also ] \ G is a free topological group (respectively, maximally almost periodic) if each G is. This notion is then combined with the theory of varieties of topological groups developed by the author. For Y_ a variety of topological groups, the ^-product of groups in J/ is defined. It is shown that the V-product, V, 1 I ^ °^ a n v-ot set {G } of groups in V_ exists, is unique and is algebraically isomorphic to the usual varietal product. It is noted that the .V-product of Hausdorff groups is not necessarily Hausdorff, but is if V_ is abelian. Each G is a quotient group of V "| f G. It is proved that the V-product of free topological groups of V. and projective topological groups of .V are of the same type. Finally it is shown that V *] f G is connected if and only if each G is connected. J a
Normality and properties related to compactness in hyperspaces
Proceedings of the American Mathematical Society, 1970
Introduction. Let X be a regular Ti topological space and 2X the space of all closed nonempty subsets of X with the finite topology [8, Definition 1.7]. In Ivanova has shown that if X is a noncompact ordinal space, then 2X is nonnormal. In this paper we give a new proof of this fact. This result is then used to show that several properties of 2X are equivalent to the compactness of X. It is not known if the normality of 2X is equivalent to the compactness of X. There are some partial results in this direction though. The paracompactness of 2X is shown to be equivalent to the compactness of X and the normality of 22 is also shown to be equivalent to the compactness of X. In the last part of the paper some properties related to the countable compactness of 2X are investigated. Notation. Because of our assumptions on X, X= { {x} :x£X} is a closed subset of 2X homeomorphic to X. The set <5n(X) = {FEX: F has at most ra points} is also closed. Furthermore, the space 2X is Hausdorff. For notation and further basic results on hyperspaces see
On Closed Subsets of Free Groups
arXiv: Group Theory, 2017
We give two examples of a finitely generated subgroup of a free group and a subset, closed in the profinite topology of a free group, such that their product is not closed in the profinite topology of a free group.
Local invariance of free topological groups
Proceedings of the Edinburgh Mathematical Society, 1986
In 1948, M. I. Graev [2] proved that the free topological group on a completely regular Hausdorff space is Hausdorff, by showing that the free group admits a certain locally invariant Hausdorff group topology. It is natural to ask if Graev's locally invariant topology is the free topological group topology. If X has the discrete topology, the answer is clearly in the affirmative. In 1973, Morris-Thompson [6] showed that if X is not totally disconnected then the answer is negative. Nickolas [7] showed that this is also the case if X has any (non-trivial) convergent sequence (for example, if X is any non-discrete metric space). Recently, Fay and Smith Thomas handled the case when X has a completely regular Hausdorff quotient space which has an infinite compact subspace (or more particularly a non-trivial convergent sequence).(Fay-Smith Thomas observe that their class of spaces includes some but not all those dealt with by Morris-Thompson.)
We introduce and study the operation, called dense amalgam, which to any tuple X_1,...,X_k of non-empty compact metric spaces associates some disconnected perfect compact metric space, denoted widetildesqcup(X1,...,Xk)\widetilde\sqcup(X_1,...,X_k)widetildesqcup(X1,...,Xk), in which there are many appropriately distributed copies of the spaces X_1,...,X_k. We then show that, in various settings, the ideal boundary of the free product of groups is homeomorphic to the dense amalgam of boundaries of the factors. We give also related more general results for graphs of groups with finite edge groups. We justify these results by referring to a convenient characterization of dense amalgams, in terms of a list of properties, which we also provide in the paper. As another application, we show that the boundary of a Coxeter group which has infinitely many ends, and which is not virtually free, is the dense amalgam of the boundaries of its maximal 1-ended special subgroups.
Topological properties of spaces admitting free group actions
Journal of Topology, 2012
In 1992, David Wright proved a remarkable theorem about which contractible open manifolds are covering spaces. He showed that if a one-ended open manifold M n has pro-monomorphic fundamental group at infinity which is not protrivial and is not stably Z, then M does not cover any manifold (except itself). In the non-manifold case, Wright's method showed that when a one-ended, simply connected, locally compact ANR X with pro-monomorphic fundamental group at infinity admits an action of Z by covering transformations then the fundamental group at infinity of X is (up to pro-isomorphism) an inverse sequence of finitely generated free groups. We improve upon this latter result, by showing that X must have a stable finitely generated free fundamental group at infinity. Simple examples show that a free group of any finite rank is possible.