Locally Compact Semigroups with Dense Maximal Subgroups

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.

Compactness and connectedness in topological groups

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.

All Hartman–Mycielski groups are minimally almost periodic

Generalized metric spaces with algebraic structures

Some aspects of dimension theory for topological groups

1. In view of the recent work on topological groups it is natural to consider the situation which arises when such groups act as transformation groups on various types of spaces. Such a study is begun here from the point of view of almost periodic transformation groups, the definition of which is suggested by von Neumann's paper on almost periodic functions in a group.t Compact topological transformation groups are a special case of almost periodic transformation groups, at least for a rather wide class of spaces. The paper concerns itself chiefly with the nature of the minimal closed invariant sets of such groups. There are some results for general spaces but the main results are for Euclidean spaces and more particularly for threedimensional Euclidean spaces. One of the most interesting theorems states that if a compact one-dimensional group acts on three-space in such a way that its orbits are uniformly bounded in diameter, then every point of the space is fixed under the group, so that if such a group is to act in a nontrivial manner the diameters of its orbits must be unbounded. Under some restrictions a similar theorem is proved for one-parameter almost periodic groups. Furthermore it is shown that for this latter class of groups, many of the orbits must actually be simple closed curves if they have one-dimensional closures. 2. The group considered here will be denoted by G. It will be subjected to various conditions as the occasion demands but it will always be Abelian. In case it is the group of real numbers, it will be spoken of as a one-parameter group; in case it is the real numbers reduced modulo one, it will be spoken of as the circle group. The space on which the group acts will be denoted by R. It will be specialized in various ways, but in any case it will always be a locally compact metric space. If x and y are two points of R, the distance between them will be denoted by d(x, y).

Totally Lindelöf and totally ω-narrow paratopological groups

In this chapter we introduce several most important notions and constructions concerning topological groups and operations with them. In particular, we develop the technique of prenorms on groups, describe in detail the construction of the Raĭkov completion of a topological group, prove a theorem on embeddings of topological groups into groups of isometries in the topology of pointwise convergence. We also introduce the class of locally compact topological groups and provide the reader with the first basic facts about groups in this class (deeper results in this direction will be presented in Chapter 9). Then we define and study the important classes of !-narrowtopological groups and of precompact (equivalently, totally bounded) topological groups. We finish this chapter with the Hartman{Mycielski construction of an embedding of an arbitrary topological group into a connected, locally pathwise connected topological group.

In this note, we firstly discuss some properties of spaces which are countably sieve-complete, densely q-complete and strongly Baire. By some known conclusions, we finally show that if G is a regular countably sieve-complete semitopological group then G is a topological group. If a regular semitopological group G has a dense subgroup which is countably sieve-complete (densely q-complete), then G is a topological group. If G is a regular countably sieve-complete semitopological group then G is a D-space if and only if G is paracompact. We point out that some conditions in Theorem 2.14 and Corollary 2.15 in [17] are not essential.

A study on weakly Lindelöf (para)topological groups with a q-point and m([formula omitted])-factorizability

This book provides a general framework for doing geometric group theory for many non-locally-compact topological transformation groups that arise in mathematical practice, including homeomorphism and diffeomorphism groups of manifolds, isometry groups of separable metric spaces and automorphism groups of countable structures. Using Roe's framework of coarse structures and spaces, the author defines a natural coarse geometric structure on all topological groups. This structure is accessible to investigation, especially in the case of Polish groups, and often has an explicit description, generalising well-known structures in familiar cases including finitely generated discrete groups, compactly generated locally compact groups and Banach spaces. In most cases, the coarse geometric structure is metrisable and may even be refined to a canonical quasimetric structure on the group. The book contains many worked examples and sufficient introductory material to be accessible to beginning graduate students. An appendix outlines several open problems in this young and rich theory.

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.

Given a T0 paratopological group G and a class C of continuous homomorphisms of paratopological groups, we define the C-semicompletionC[G) and C-completionC[G] of the group G that contain G as a dense subgroup, satisfy the T0-separation axiom and have certain universality properties. For special classes C, we present some necessary and sufficient conditions on G in order that the (semi)completions C[G) and C[G] be Hausdorff. Also, we give an example of a Hausdorff paratopological abelian group G whose C-semicompletion C[G) fails to be a T1-space, where C is the class of continuous homomorphisms of sequentially compact topological groups to paratopological groups. In particular, the group G contains an ω-bounded sequentially compact subgroup H such that H is a topological group but its closure in G fails to be a subgroup.

Fréchet–Urysohn subspaces of free topological groups III