Compactness and connectedness in topological groups

1. Introduction.Let S be a locally compact semigroup with a dense maximal subgroup G.We ask the following question: When is G a topological group under the induced topology?It was proved recently by P. S. Mostert [8] that in general G is not a topological group.However, there are several good reasons motivating the study of such a problem.First, for an arbitrary locally compact semigroup, the closure of the maximal subgroup containing an idempotent forms a locally compact semigroup.If we know this maximal subgroup is a topological group, then it is a locally compact topological group.(This fact will be proved in the following paragraph.)By our rather good knowledge of locally compact topological groups, we can obtain information about the structure of this locally compact semigroup.Second, if we look at this problem from the point of view of transformation groups, we have a group G acting on a locally compact topological space.Naturally, we would like this group to be topologized so that it is a topological group.It is known that the weakest topology for G to be a topological group is the g-topology [1].It is somewhat harder to deal with the g-topology than with the fc-topology, but under some conditions the ^-topology will coincide with the fe-topology.Among the known conditions, the local compactness of the group seems most interesting.If one tries to generalize such a condition, it is rather natural to ask the following : If G is a group of homeomorphisms of X onto X, and the closure of G is locally compact in the fc-topology, does the induced topology of G coincide with the g-topology?Third, in the past few years there have been several papers about such questions : What is the relationship between the algebraic structure and the topological structure?When can one strengthen the topological structure so that even under weak conditions the algebraic operations will be continuous in a certain sense?For instance, in topological groups, if the group space is locally compact, and if the multiplication is separately continuous with respect to the topology of the space, then the group is a topological group.We are able to prove the same theorem when the space is second countable and second category without the assumption of local compactness [9].Now, our present problem arises quite

All Hartman–Mycielski groups are minimally almost periodic

A now standard technique for treating a class of Baire set problems, originating with Halmos [2; 51.D], is to construct a metric space that, in a way, "fixes" a given Baire set. Formally this means, given a Baire set B in a space X, find a metric space M and a map g of X onto M with the property that B = g l gB . In phrasing the problem for topological groups it is natural to require that M be a group and g be a homomorphism. Specifically, what is desired is a subgroup H that "fixes" B, i.e., such that B = BH. This problem is solved in Halmos for locally compact a-compact groups in a theorem [2; 64.G] that has led to interesting consequences elsewhere [4, 1]. As will be seen, Halmos' theorem is also true for totally bounded groups. An example of Sirota's of a countable topological group with no non-trivial metrizable homomorphic images indicates that, in this direction at least, the barrier of local compactness cannot be lifted without giving up metrizability in M. If this is done, however, replacing the requirement that M be metrizable, i.e., that H have a countable base in G, with the less stringent requirement that H be a G~-set in G, an analogue of Halmos' theorem is true for Lindelff groups. In view of these remarks it is convenient to distinguish between two distinct Baire set fixing problems. A closed invariant subgroup H of a group G is said to be a fixiru3 subgroup (M-fixing subgroup) provided H is a G~-set (H has a countable base of neighborhoods in G). G satisfies the Baire set fixing property (M-Baire set fixing property) provided, for each Baire set in G, there is a fixing subgroup (M-fixing subgroup) such that B = BH. The Baire sets are taken to be the elements of the smallest a-algebra containing the zero sets of continuous real-valued functions on G. For technical convenience in describing proofs, a G~-set containing the identity is called an admissible set.

Let $Q \in$ character $\leq \tau$, pseudocharacter $\leq \tau$, tightness $\leq \tau$, weight $\leq \tau$ , ${P_\tau }$-property, discreteness, Fréchet-Urysohn property, sequentiality, radiality, pseudoradiality, local compactness, $k$-property. If ${X^n} = \cup \{ {X_i}:i \in n\}$, ${X_i} \vdash Q$ for all $i \in n$ then $X \vdash Q$ (i.e. the property $Q$ is $n$-additive in ${X^n}$ for any $X \in {T_3}$). Metrizability is $n$-additive in ${X^n}$ provided $X$ is compact or $c(X) = \omega$. ${\text {ANR}}$-property is closely $n$-additive in ${X^n}$ if $X$ is compact ("closely" means additivity in case ${X_i}$ is closed in ${X^n}$). If $Q \in$ metrizability, character $\leq \tau$, pseudocharacter $\leq \tau$, diagonal number $\leq \tau$ , $i$-weight $\leq \tau$, pseudoweight $\leq \tau$, local compactness then $Q$ is finitely additive in any topological group.

Let G be a reflexive topological group, and Gits group of characters, endowed with the compact open topology. We prove that the evaluation mapping from Gx G into the torus T is continuous if and only if G is locally compact. This is an analogue of a well-known theorem of Arens on admissible topologies on C(X) . DEFINITIONS AND REMARKS Let X, Y be topological spaces, and let Z be a subset of yX . A topology on Z is said to be admissible if the evaluation mapping from the product Z x X into Y, defined by w(f, x) = f(x), is continuous. Let (S, V) = {f E Z; f(S) C V}. The family {(S, V)}, where S runs over the collection of all compact subsets of X and V runs over a basis of open sets in Y, is a subbase for the compact open topology on Z. An admissible topology on Z must be finer than the compact open topology [8]. A result of Arens states that the existence of a coarsest admissible topology for the class of real continuous functions on a completely regular space X is equivalent to X being locally compact [1]. In this paper we are interested in reflexive topological groups. Answering in the negative a question of Megrelishvili [7], we will see that the evaluation map for those groups need not be continuous. In fact, as specified in the theorem, the continuity of the evaluation characterizes locally compact Hausdorff abelian groups among reflexive groups. We prove this result by means of convergence spaces. For an account of theory of convergence spaces the reader is referred to [3, 4]. We only give here needed definitions. Let G be a Hausdorff topological abelian group, and let FG be the set of all continuous homomorphisms from G into the unidimensional torus T. If addition is defined pointwise in FG, it becomes an abelian group. The continuous Received by the editors October 5, 1993 and, in revised form, April 25, 1994; the contents of this paper were presented to the First Ibero-American Conference on Topology and its Applications in Benicassim, Spain, on March 28-30, 1995. 1991 Mathematics Subject Classification. Primary 22A05.

A class of abelian topological groups was previously defined to be a variety of topological groups with coproducts if it is closed under forming subgroups, quotients, products and coproducts in the category of all abelian topological groups and continuous homomorphisms. This extended research on varieties of topological groups initiated by the second author. The key to describing varieties of topological groups generated by various classes was proving that all topological groups in the variety are a quotient of a subgroup of a product of groups in the generating class. This paper analyses generating varieties of topological groups with coproducts. It focuses on the interplay between forming products and coproducts. It is proved that the variety of topological groups with coproducts generated by all discrete groups contains topological groups which cannot be expressed as a quotient of a subgroup of a product of a coproduct of discrete groups. It is proved that the variety of topological groups with coproducts generated by any infinite-dimensional Hilbert space contains all infinite-dimensional Hilbert spaces, answering an open question. This contrasts with the result that a variety of topological groups generated by a topological group does not contain any infinite-dimensional Hilbert space of greater cardinality.

A. R. Bernstein introduced the concept of p‐compactness (p∈ω*). Years later, W. W. Comfort defined a (pre)‐order ≤C on ω* by p ≤C q iff every q‐compact space is p‐compact. It is natural to consider the relation ≤GC on ω* given by p≤GC q iff every q‐compact topological group is p‐compact. It is shown that ≤RK≤C, and ≤GC coincide on the set of weak P‐points of ω* (≤RK denotes the Rudin–Keisler order on ω*), and that ≤C and ≤GC have several properties in common, but the question whether ≤C=≤GC still remains unsolved. Some properties of free M‐compact groups (M⊆α*) are also studied. In particular, it is shown that if α is a regular infinite cardinal and X is < α‐bounded, then X is a closed subspace of the free < α‐bounded topological (Abelian) group over X.

1. Let (X, T, 7r) be a topological transformation group, where X is a nontrivial Hausdorff space and T is a topological group which leaves an end point e of X fixed. In [3], Wallace proved that if T is cyclic and X is a locally connected continuum, T has a fixed point other than e. Then in [4], Wallace asked the following question: If X is a peano continuum and T is compact or abelian, then does T have a fixed point other than e? In [5] Wang showed that if T is compact, and X is arcwise connected, then T has a fixed point other than e. Then Chu [1 ] showed that T has infinitely many fixed points. Chu began the study of the abelian case in [2]. In this paper we show by example that X may be a peano continuum and T may be a countably generated abelian group which has e for its only fixed point, ?2. Thus in general the answer to Wallace's question in the abelian case is no. However, if T is a generative group, and X is compact and arcwise connected, then T has a fixed point other than e, ?3. We also show by example that T may be a finitely generated nonabelian group which has e for its only fixed point, ?4.

On the discontinuity of the π1-action

Let (X,+) be a topological abelian group. We discuss regularity of solutions f:X→R of Hlawka’s functional inequality f(x+y)+f(y+z)+f(x+z)≤f(x+y+z)+f(x)+f(y)+f(z), postulated for all x,y,z∈X. We study the lower and upper hull of f. Moreover, we provide conditions which imply continuity of f. We prove, in particular, that if X is generated by any neighborhood of zero, f is continuous at zero, and f(0)=0, then f is continuous on X.

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

As an effective method to enhance the dielectric performance of polyolefin materials, polar side group modification has been extensively applied in the insulation and energy storage materials of electrical and electronic systems. In this work, two side groups with different topological structures were adopted, namely, vinyl acetate (VAc, aliphatic chain) and N-vinyl-pyrrolidone (NVP, saturated ring), to modify polypropylene (PP) via chemical grafting, and the effects of structural topology of the polar side group on the microscopic and macroscopic characteristics of PP, particularly on its electrical anti-breakdown ability, were investigated. Experimental results showed that the side group structural topology directly affected the crystallization and thermal properties of PP. The in-depth computational analysis indicated that the grafted NVP possessed a lower deep trap depth than VAc, which is related to the topological structure and corresponding orbital interaction within the side group. Furthermore, molecular dynamic (MD) simulations revealed the presence of a saturated ring in the NVP side group that led to more free volume within the material's condensed state than VAc. Therefore, by contrast, VAc-grafted PP with deeper trap orbitals and less free volume exhibited higher breakdown strength enhancement up to 21% and 14% at 30 and 90 °C, respectively. Thus, this work provides a novel understanding of the topological structure effect of the side group on the macroscopic dielectric performance from the viewpoint of microscopic physical chemistry. Furthermore, this work would serve as a reference for the refined design and property modulation of dielectric materials in modern electrical power facilities.

Weakly complete free topological groups

The survey examines the properties of the Tikhonov space X that depend on the topological properties of the function space C(X) with pointwise topology and of the free topological group F(X). It covers the part of the theory of topological groups dealing with topological invariants and topological classification of topological groups. Continuous images of topological groups are considered in the context of studying the actions of topological groups on spaces.

The classifying space BG of a topological group G can be filtered by a sequence of subspaces B(q,G) using the descending central series of free groups. If G is finite, describing them as homotopy colimits is convenient when applying homotopy theoretic methods. In this thesis we introduce natural subspaces B(q,G)_p of B(q,G) defined for a fixed prime p. We show that B(q,G) is stably homotopy equivalent to a wedge sum of B(q,G)_p as p runs over the primes dividing the order of G. Colimits of abelian groups play an important role in understanding the homotopy type of these spaces. Extraspecial p-groups are key examples, for which these colimits turn out to be finite. We prove that for extraspecial p-groups of rank at least 4 the space B(2,G) does not have the homotopy type of a K(π,1) space. Furthermore, we give a group theoretic condition, applicable to symmetric groups and general linear groups, which implies the space B(2,G) not having the homotopy type of a K(π,1) space. For a finite group G, we compute the complex K-theory of B(2,G) modulo torsion.