Sequential criteria for equicontinuity and uniformities on topological groups

We prove that a compact space embeds into a ‐product of compact metrizable spaces (‐product of intervals) if and only if is (strongly countable‐dimensional) hereditarily metalindelöf and every subspace of has a nonempty relative open second countable subset. This provides novel characterizations of ‐Corson and compact spaces. We give an example of a uniform Eberlein compact space that does not embed into a product of compact metric spaces in such a way that the ‐product is dense in the image. In particular, this answers a question of Kubiś and Leiderman. We also show that for a compact space , the property of being compact is determined by the topological structure of the space of continuous real‐valued functions of equipped with the pointwise convergence topology. This refines a recent result of Zakrzewski.

First, we discuss the behavior of boundedness in uniform spaces with respect to subspaces, projective limits, and suprema in relation to precompactness. A special uniformly isomorphic embedding of un arbitrary uniform space in a bounded uniform space is presented and examined in 2.6. Hejcman’s characterization (by B-conservativity) of uniform spaces in which boundedness can be tested by a single pseudometric is proved in a new way, see 3.13, using a version 3.1 of the metrization lemma. We comment briefly on boundedness in topological vector spaces. In topological groups we investigate a hierarchy of partly new notions of boundedness, strongly interrelated among themselves, and exhibit various situations in which certain of these notions coincide. Boundedness respecting subspaces of a uniform space prove useful. Many examples illustrate and complement the general theory, see, e.g., Example 6.4.

Incomplete infima of complete uniformities related to topological groups and topological vector spaces

Some aspects of dimension theory for topological groups

All Hartman–Mycielski groups are minimally almost periodic

A density theorem of semisimple analytic groups acting on locally compact groups is presented.Let G and H be locally compact groups with G acting continuously on H as a group of automorphisms.An element h of H is said to be G-bounded if the orbit Gh = \g(h): g G] has compact closure in H.We write F AH) fot the set of all G-bounded elements in H.It is very easy to verify that F AH) is a G-invariant subgroup of H. However in general, EAH) is not closed in H.In this paper, we shall study the group EAH) fot certain topological groups G. Our main result is the following Theorem.Let G be a semisimple analytic group without compact actors acting on a locally compact group H continuously as a group of automorphisms.If the set F AH) is dense in H, then G acts trivially on H.The theorem generalizes some results in [2], [4] and is closely related to the density property of certain subgroups in semisimple analytic groups without compact factors.The result of Corollaries 4.1 and 4.2 is contained in [2], [4].In the sequel, we shall use the term "G acts on H" fot "G acts on H as a group of automorphisms".1. Minimally almost periodic groups.Let G be a locally compact group.We recall that G is minimally almost periodic if there are no nontrivial continuous homomorphisms /: G -* G of locally compact groups such that the closure C1(/(G)) of (G) in G is compact.Minimally almost periodic groups have been widely studied.Yet for our need, we shall establish some lemmas concerning minimally almost periodic groups.Lemma 1.1.Let G be a minimally almost periodic group acting continuously on a locally compact group H, and N be a closed G-invariant normal subgroup of H.

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.

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.

For a domain  in C, we denote by Aut() the group of holomorphic automorphisms of . It is obvious that Aut() is a topological group with respect to the law of composition and the compact-open topology. In particular, it is a theorem of H. Cartan that Aut() is in fact a Lie group if  is bounded. In light of the outstanding question “Which domains possess noncompact automorphism group?” there is much interest focused upon the existence and nonexistence of orbits of the automorphism group action accumulating at a given boundary point. The well-known Greene–Krantz conjecture belongs to such a line of research. In this paper, we discuss the finite-type boundary points that repel automorphism orbits. Denote by τ6(q) the D’Angelo type (see [10]) at q of the real hypersurface 6 in C. Now we present our main theorem.

1. Let G be a compact group, let I b e a subset of its dual object 𝚪, which, without loss of generality, will be assumed to be a countable subset. Let Di, i ∈ I , be irreducible representations of G of degree di. The Fourier series of a function F in L1(G) is denned bywhere

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

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.

The purpose of this paper is to introduce the concept of soft uniform spaces and the relationships between soft uniform spaces and uniform spaces. The notions of soft uniform structure, soft uniform continious function, and operations on soft uniform space are introduced and their basic properties are investigated.