Is it true that the left and the right uniformities on a topological group coincide as soon as every left uniformly continuous real valued function is right uniformly continuous? This question is known as Itzkowitz's problem, and it is still open. We show that the generalization of the problem to homogeneous factor-spaces of topological groups has a negative answer.
In this paper, we deal with uniform spaces whose diagonal uniformity admits a basis consisting of equivalence relations. Such non-Archimedean uniform spaces are particularly interesting for applications in commutative ring theory, because uniformities stemming from valuations or directed systems of ideals are of this type. In general, apart from diagonal uniformities, there are two further approaches to the concept of uniform spaces: covering uniformities and systems of pseudo-metrics. For each of these ways of defining a uniformity, we isolate a non-Archimedean special case and show that thes…
The answer about Itzkowitz's Problems on FSIN groups
Institute of Mathematics, Nanjing Normal University（南京师范大学数学系）；School of Mathematics and Computational Science, Wuyi University（五邑大学数学与计算科学学院）；School of Mathematical Sciences, Guangxi Minzu University（广西民族大学数学科学学院）
In this paper, we introduce the notions of pre-uniform spaces and pre-proximities and investigate some basic properties about them, where the definition of pre-uniformity here is different with the pre-uniformities which are studied in \cite{BR2016}, \cite{GM2007} and \cite{K2016} respectively. First, we prove that each pre-uniform pre-topology is regular, and give an example to show that there exists a pre-uniform structure on a finite set such that the pre-uniform pre-topology is not discrete. Moreover, we give three methods of generating (strongly) pre-uniformities, that is, the definition …
Two classes of topological spaces are introduced on which every probability Radon measure possesses a uniformly distributed sequence or a uniformly tight uniformly distributed sequence. It is shown that these classes are stable under multiplication by completely regular Souslin spaces
A topological group is minimal if it does not admit a strictly coarser Hausdorff group topology. The Roelcke uniformity (or lower uniformity) on a topological group is the greatest lower bound of the left and right uniformities. A group is Roelcke-precompact if it is precompact with respect to the Roelcke uniformity. Many naturally arising non-Abelian topological groups are Roelcke-precompact and hence have a natural compactification. We use such compactifications to prove that some groups of isometries are minimal. In particular, if U_1 is the Urysohn universal metric space of diameter 1, the…
We provide a representation of the homomorphisms $U\longrightarrow \mathbb R$, where $U$ is the lattice of all uniformly continuous on the line. The resulting picture is sharp enough to describe the fine topological structure of the space of such homomorphisms.
For any topological space $X$ we study the relation between the universal uniformity $\mathcal U_X$, the universal quasi-uniformity $q\mathcal U_X$ and the universal pre-uniformity $p\mathcal U_X$ on $X$. For a pre-uniformity $\mathcal U$ on a set $X$ and a word $v$ in the two-letter alphabet $\{+,-\}$ we define the verbal power $\mathcal U^v$ of $\mathcal U$ and study its boundedness numbers $\ell(\mathcal U^v)$ and $\bar \ell(\mathcal U^v)$. The boundedness numbers of the (Boolean operations over) the verbal powers of the canonical pre-uniformities $p\mathcal U_X$, $q\mathcal U_X$ and $\math…
We study free topological groups defined over uniform spaces in some subclasses of the class NA of non-archimedean groups. Our descriptions of the corresponding topologies show that for metrizable uniformities the corresponding free balanced, free abelian and free Boolean NA groups are also metrizable. Graev type ultra-metrics determine the corresponding free topologies. Such results are in a striking contrast with free balanced and free abelian topological groups cases (in standard varieties). Another contrasting advantage is that the induced topological group actions on free abelian NA group…
We show that the topology of uniform convergence on bounded sets is compatible with the group law of the automorphism group of a large class of spaces that are endowed with both a uniform structure and a bornology, thus yielding numerous examples of topological groups. This encompasses many of previously known cases, such as homeomorphism groups of locally compact spaces, automorphism groups of first-order structures, general linear groups of normed spaces, automorphism groups of uniform spaces, etc. We also study the automorphism group of a topological group within this framework (generalizin…
The notion of locally quasi-convex abelian group, introduce by Vilenkin, is extended to maximally almost-periodic non-necessarily abelian groups. For that purpose, we look at certain bornologies that can be defined on the set $\hbox{rep}(G)$ of all finite dimensional continuous representations on a topological group $G$ in order to associate well behaved group topologies (dual topologies) to them. As a consequence, the lattice of all Hausdorff totally bounded group topologies on a group $G$ is shown to be isomorphic to the lattice of certain special subsets of $\hbox{rep}(G_d)$. Moreover, gene…
Countable uniformly discrete sets in functionally balanced groups
Institute of Mathematics, Nanjing Normal University（南京师范大学数学研究所）；School of Mathematics and Computational Science, Wuyi University（五邑大学数学与 computational 科学学院）；School of Mathematical Sciences, Guangxi Minzu University（广西民族大学数学科学学院）
Uniform measures are the functionals on the space of bounded uniformly continuous functions that are continuous on every bounded uniformly equicontinuous set. This paper describes the role of uniform measures in the study of convolution on an arbitrary topological group.
A topological group is said to be ambitable if each uniformly bounded uniformly equicontinuous set of functions on the group with its right uniformity is contained in an ambit. For n=0,1,2,..., every locally aleph_n bounded topological group is either precompact or ambitable. In the familiar semigroups constructed over ambitable groups, topological centres have an effective characterization.
Let $X$ be a locally compact topological space, $(Y,d)$ be a boundedly compact metric space and $LB(X,Y)$ be the space of all locally bounded functions from $X$ to $Y$. We characterize compact sets in $LB(X,Y)$ equipped with the topology of uniform convergence on compacta. From our results we obtain the following interesting facts for $X$ compact: $\bullet$ If $(f_n)_n$ is a sequence of uniformly bounded finitely equicontinuous functions of Baire class $α$ from $X$ to $\Bbb R$, then there is a uniformly convergent subsequence $(f_{n_k})_k$; $\bullet$ If $(f_n)_n$ is a sequence of uniformly bou…
Every topological group $G$ has some natural compactifications which can be a useful tool of studying $G$. We discuss the following constructions: (1) the greatest ambit $S(G)$ is the compactification corresponding to the algebra of all right uniformly continuous bounded functions on $G$; (2) the Roelcke compactification $R(G)$ corresponds to the algebra of functions which are both left and right uniformly continuous; (3) the weakly almost periodic compactification $W(G)$ is the envelopping compact semitopological semigroup of $G$ (`semitopological' means that the multiplication is separately …
According to Kat\vetov (1988), for every infinite cardinal $\mathfrak m$ satisfying ${\mathfrak m}^{\mathfrak n}\leq {\mathfrak m}$ for all ${\mathfrak n}<{\mathfrak m}$, there exists a unique $\mathfrak m$-homogeneous universal metric space $\Ur_{\mathfrak m}$ of weight $\mathfrak m$. This object generalizes the classical Urysohn universal metric space $\Ur = \Ur_{\aleph_0}$. We show that for $\mathfrak m$ uncountable, the isometry group $\Iso(\Urm)$ with the topology of simple convergence is not a universal group of weight $\mathfrak m$: for instance, it does not contain $\Iso(\Ur)$ as a top…
We continue in this paper the study of locally minimal groups started in \cite{LocMin}. The minimality criterion for dense subgroups of compact groups is extended to local minimality. Using this criterion we characterize the compact abelian groups containing dense countable locally minimal subgroups, as well as those containing dense locally minimal subgroups of countable free-rank. We also characterize the compact abelian groups whose torsion part is dense and locally minimal. We call a topological group $G$ {\it almost minimal} if it has a closed, minimal normal subgroup $N$ such that the qu…
In spaces of metrics, we investigate topological distributions of the doubling property, the uniform disconnectedness, and the uniform perfectness, which are the quasi-symmetrically invariant properties appearing in the David--Semmes theorem. We show that the set of all doubling metrics and the set of all uniformly disconnected metrics are dense in spaces of metrics on finite-dimensional and zero-dimensional compact metrizable spaces, respectively. Conversely, this denseness of the sets implies the finite-dimensionality, zero-dimensionality, and the compactness of metrizable spaces. We also de…
A subset $A$ of $\nats$ is called an IP-set if $A$ contains all finite sums of distinct terms of some infinite sequence $(x_n)_{n\in \nats} $ of natural numbers. Central sets, first introduced by Furstenberg using notions from topological dynamics, constitute a special class of IP-sets possessing rich combinatorial properties: Each central set contains arbitrarily long arithmetic progressions, and solutions to all partition regular systems of homogeneous linear equations. In this paper we investigate central sets in the framework of combinatorics on words. Using various families of uniformly r…
A uniform space $X$ is said to be proximally fine if every proximally continuous map on $X$ into a uniform is uniformly continuous. We supply a proof that every topological group which is functionnaly generated by its precompact subsets is proximally fine with respect to its right uniformity. On the other hand, we show that there are various permutation groups $G$ on the integers $\mathbb N$ that are not proximally fine with respect to the topology generated by the sets $\{g\in G: g(A)\subset B\}$, $A,B\subset \mathbb N$.
Itzkowitz's problem asks whether every topological group $G$ has equal left and right uniform structures provided that bounded left uniformly continuous real-valued function on $G$ are right uniformly continuous. This paper provides a positive answer to this problem if $G$ is of bounded exponent or, more generally, if there exist an integer $p\geq 2$ and a nonempty open set $U\subset G$ such that the power map $U\ni g\to g^p\in G$ is left (or right) uniformly continuous. This also resolves the problem for periodic groups which are Baire spaces.
We prove that the additive group $(E^\ast,τ_k(E))$ of an $\mathscr{L}_\infty$-Banach space $E$, with the topology $τ_k(E)$ of uniform convergence on compact subsets of $E$, is topologically isomorphic to a subgroup of the unitary group of some Hilbert space (is \emph{unitarily representable}). This is the same as proving that the topological group $(E^\ast,τ_k(E))$ is uniformly homeomorphic to a subset of $\ell_2^κ$ for some $κ$. As an immediate consequence, preduals of commutative von Neumann algebras or duals of commutative $C^\ast$-algebras are unitarily representable in the topology of uni…
In this paper, we first evaluate topological distributions of the sets of all doubling spaces, all uniformly disconnected spaces, and all uniformly perfect spaces in the space of all isometry classes of compact metric spaces equipped with the Gromov--Hausdorff distance. We then construct branching geodesics of the Gromov--Hausdorff distance continuously parameterized by the Hilbert cube, passing through or avoiding sets of all spaces satisfying some of the three properties shown above, and passing through the sets of all infinite-dimensional spaces and the set of all Cantor metric spaces. Our …
Let $C(\mathbf I)$ be the set of all continuous self-maps from ${\mathbf I}=[0,1]$ with the topology of uniformly convergence. A map $f\in C({\mathbf I})$ is called a transitive map if for every pair of non-empty open sets $U,V$ in $\mathbf{I}$, there exists a positive integer $n$ such that $U\cap f^{-n}(V)\not=\emptyset.$ We note $T(\mathbf{I})$ and $\overline{T(\mathbf{I})}$ to be the sets of all transitive maps and its closure in the space $C(\mathbf I)$. In this paper, we show that $T(\mathbf{I})$ and $\overline{T(\mathbf{I})}$ are homeomorphic to the separable Hilbert space $\ell_2$.
