Homeomorphism and Isomorphism of Abelian Groups

1974 ◽  
Vol 26 (6) ◽  
pp. 1515-1519 ◽  
Author(s):  
Stephen Scheinberg
Keyword(s):  
Topological Group ◽  
Abelian Group ◽  
Topological Space ◽  
Abelian Groups ◽  
Group Structure ◽  
Locally Compact ◽  
Abelian Topological Group

An abelian topological group can be considered simply as an abelian group or as a topological space. The question considered in this article is whether the topological group structure is determined by these weaker structures. Denote homeomorphism, isomorphism, and homeomorphic isomorphism by ≈, ≅ , and =, respectively. The principal results are these.Theorem 1. If G1andG2are locally compact and connected, then G1≈ G2implies G1= G2.

scholarly journals On orderability of topological groups

1985 ◽  
Vol 8 (4) ◽  
pp. 747-754
Author(s):  
G. Rangan
Keyword(s):  
Topological Group ◽  
Abelian Group ◽  
Group Structure ◽  
Locally Compact Abelian Group ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Topological Groups ◽  
Locally Compact ◽  
Necessary And Sufficient ◽  
Totally Disconnected

A necessary and sufficient condition for a topological group whose topology can be induced by a total order compatible with the group structure is given and such groups are called ordered or orderable topological groups. A separable totally disconnected ordered topological group is proved to be non-archimedean metrizable while the converse is shown to be false by means of an example. A necessary and sufficient condition for a no-totally disconnected locally compact abelian group to be orderable is also given.

scholarly journals Commutative Topological Semigroups Embedded into Topological Abelian Groups

Axioms ◽  
2020 ◽  
Vol 9 (3) ◽  
pp. 87
Author(s):  
Julio César Hernández Arzusa
Keyword(s):  
Topological Group ◽  
Abelian Group ◽  
Topological Semigroup ◽  
Abelian Groups ◽  
Sufficient Conditions ◽  
Locally Compact ◽  
Topological Abelian Group ◽  
Topological Semigroups ◽  
Topological Monoid

In this paper, we give conditions under which a commutative topological semigroup can be embedded algebraically and topologically into a compact topological Abelian group. We prove that every feebly compact regular first countable cancellative commutative topological semigroup with open shifts is a topological group, as well as every connected locally compact Hausdorff cancellative commutative topological monoid with open shifts. Finally, we use these results to give sufficient conditions on a commutative topological semigroup that guarantee it to have countable cellularity.

scholarly journals CONTINUITY OF MEASURABLE HOMOMORPHISMS

2008 ◽  
Vol 78 (1) ◽  
pp. 171-176 ◽  
Author(s):  
JANUSZ BRZDȨK
Keyword(s):  
Topological Group ◽  
Abelian Group ◽  
Baire Property ◽  
Topological Groups ◽  
Locally Compact ◽  
Compact Topological Group ◽  
Locally Compact Topological Group

AbstractWe give some general results concerning continuity of measurable homomorphisms of topological groups. As a consequence we show that a Christensen measurable homomorphism of a Polish abelian group into a locally compact topological group is continuous. We also obtain similar results for the universally measurable homomorphisms and the homomorphisms that have the Baire property.

scholarly journals Convergence of ergodic averages for many group rotations

2015 ◽  
Vol 36 (7) ◽  
pp. 2107-2120
Author(s):  
ZOLTÁN BUCZOLICH ◽  
GABRIELLA KESZTHELYI
Keyword(s):  
Topological Group ◽  
Abelian Group ◽  
Measurable Function ◽  
Dual Group ◽  
Ergodic Averages ◽  
Abelian Topological Group ◽  
Compact Abelian Topological Group ◽  
Image Position ◽  
Rotation Set ◽  
Group Rotations

Suppose that $G$ is a compact Abelian topological group, $m$ is the Haar measure on $G$ and $f:G\rightarrow \mathbb{R}$ is a measurable function. Given $(n_{k})$, a strictly monotone increasing sequence of integers, we consider the non-conventional ergodic/Birkhoff averages $$\begin{eqnarray}M_{N}^{\unicode[STIX]{x1D6FC}}f(x)=\frac{1}{N+1}\mathop{\sum }_{k=0}^{N}f(x+n_{k}\unicode[STIX]{x1D6FC}).\end{eqnarray}$$ The $f$-rotation set is $$\begin{eqnarray}\unicode[STIX]{x1D6E4}_{f}=\{\unicode[STIX]{x1D6FC}\in G:M_{N}^{\unicode[STIX]{x1D6FC}}f(x)\text{ converges for }m\text{ almost every }x\text{ as }N\rightarrow \infty \}.\end{eqnarray}$$We prove that if $G$ is a compact locally connected Abelian group and $f:G\rightarrow \mathbb{R}$ is a measurable function then from $m(\unicode[STIX]{x1D6E4}_{f})>0$ it follows that $f\in L^{1}(G)$. A similar result is established for ordinary Birkhoff averages if $G=Z_{p}$, the group of $p$-adic integers. However, if the dual group, $\widehat{G}$, contains ‘infinitely many multiple torsion’ then such results do not hold if one considers non-conventional Birkhoff averages along ergodic sequences. What really matters in our results is the boundedness of the tail, $f(x+n_{k}\unicode[STIX]{x1D6FC})/k$, $k=1,\ldots ,$ for almost every $x$ for many $\unicode[STIX]{x1D6FC}$; hence, some of our theorems are stated by using instead of $\unicode[STIX]{x1D6E4}_{f}$ slightly larger sets, denoted by $\unicode[STIX]{x1D6E4}_{f,b}$.

The Meet Operator in the Lattice of Group Topologies

1986 ◽  
Vol 29 (4) ◽  
pp. 478-481
Author(s):  
Bradd Clark ◽  
Victor Schneider
Keyword(s):  
Topological Group ◽  
Abelian Group ◽  
Locally Compact Group ◽  
Group Topology ◽  
Locally Compact ◽  
Hausdorff Topology ◽  
Lattice Of Topologies ◽  
Group Form ◽  
Complemented Lattice ◽  
Infinite Abelian Group

AbstractIt is well known that the lattice of topologies on a set forms a complete complemented lattice. The set of topologies which make G into a topological group form a complete lattice L(G) which is not a sublattice of the lattice of all topologies on G.Let G be an infinite abelian group. No nontrivial Hausdorff topology in L(G) has a complement in L(G). If τ1 and τ2 are locally compact topologies then τ1Λτ2 is also a locally compact group topology. The situation when G is nonabelian is also considered.

SUBSPACES OF THE FREE TOPOLOGICAL VECTOR SPACE ON THE UNIT INTERVAL

2017 ◽  
Vol 97 (1) ◽  
pp. 110-118 ◽  
Author(s):  
SAAK S. GABRIYELYAN ◽  
SIDNEY A. MORRIS
Keyword(s):  
Topological Group ◽  
Vector Space ◽  
Topological Vector Space ◽  
Unit Interval ◽  
Vector Subspace ◽  
Locally Compact ◽  
Tychonoff Space ◽  
Abelian Topological Group ◽  
Free Abelian Topological Group ◽  
Usual Topology

For a Tychonoff space $X$, let $\mathbb{V}(X)$ be the free topological vector space over $X$, $A(X)$ the free abelian topological group over $X$ and $\mathbb{I}$ the unit interval with its usual topology. It is proved here that if $X$ is a subspace of $\mathbb{I}$, then the following are equivalent: $\mathbb{V}(X)$ can be embedded in $\mathbb{V}(\mathbb{I})$ as a topological vector subspace; $A(X)$ can be embedded in $A(\mathbb{I})$ as a topological subgroup; $X$ is locally compact.

Cohomology with coefficients in symmetric cat-groups. An extension of Eilenberg–MacLane's classification theorem

1993 ◽  
Vol 114 (1) ◽  
pp. 163-189 ◽  
Author(s):  
M. Bullejos ◽  
P. Carrasco ◽  
A. M. Cegarra
Keyword(s):  
Abelian Group ◽  
Homotopy Type ◽  
Abelian Groups ◽  
Group Structure ◽  
Cohomology Theory ◽  
Classification Theorem ◽  
Topological Spaces ◽  
Simplicial Sets

AbstractIn this paper we use Takeuchy–Ulbrich's cohomology of complexes of categories with abelian group structure to introduce a cohomology theory for simplicial sets, or topological spaces, with coefficients in symmetric cat-groups . This cohomology is the usual one when abelian groups are taken as coefficients, and the main topological significance of this cohomology is the fact that it is equivalent to the reduced cohomology theory defined by a Ω-spectrum, {}, canonically associated to . We use the spaces to prove that symmetric cat-groups model all homotopy type of spaces X with Πi(X) = 0 for all i ╪ n, n + 1 and n ≥ 3, and then we extend Eilenberg–MacLane's classification theorem to those spaces: .

scholarly journals Groups – Additive Notation

2015 ◽  
Vol 23 (2) ◽  
pp. 127-160 ◽  
Author(s):  
Roland Coghetto
Keyword(s):  
Topological Group ◽  
Group Theory ◽  
Abelian Group ◽  
Normal Subgroup ◽  
Dense Subset ◽  
Abelian Topological Group ◽  
Covering Group ◽  
Mizar Mathematical Library ◽  
Order Of An Element ◽  
Generic Theory

Abstract We translate the articles covering group theory already available in the Mizar Mathematical Library from multiplicative into additive notation. We adapt the works of Wojciech A. Trybulec [41, 42, 43] and Artur Korniłowicz [25]. In particular, these authors have defined the notions of group, abelian group, power of an element of a group, order of a group and order of an element, subgroup, coset of a subgroup, index of a subgroup, conjugation, normal subgroup, topological group, dense subset and basis of a topological group. Lagrange’s theorem and some other theorems concerning these notions [9, 24, 22] are presented. Note that “The term ℤ-module is simply another name for an additive abelian group” [27]. We take an approach different than that used by Futa et al. [21] to use in a future article the results obtained by Artur Korniłowicz [25]. Indeed, Hölzl et al. showed that it was possible to build “a generic theory of limits based on filters” in Isabelle/HOL [23, 10]. Our goal is to define the convergence of a sequence and the convergence of a series in an abelian topological group [11] using the notion of filters.

scholarly journals Embeddings into monothetic groups

2018 ◽  
Vol 21 (4) ◽  
pp. 579-581
Author(s):  
Michal Doucha
Keyword(s):  
Topological Group ◽  
Abelian Group ◽  
Elementary Proof ◽  
Invariant Metric ◽  
Abelian Topological Group

AbstractWe provide a very short elementary proof that every separable abelian group with a bounded invariant metric isometrically embeds into a monothetic group with a bounded invariant metric, in such a way that the result of Morris and Pestov that every separable abelian topological group embeds into a monothetic group is an immediate corollary. We show that the boundedness assumption cannot be dropped.

Varieties of topological groups III

1970 ◽  
Vol 2 (2) ◽  
pp. 165-178 ◽  
Author(s):  
Sidney A. Morris
Keyword(s):  
Topological Group ◽  
Abelian Group ◽  
Algebraic Variety ◽  
Additive Group ◽  
Topological Groups ◽  
Locally Compact ◽  
Circle Group ◽  
Free Topological Group ◽  
The Family ◽  
Usual Topology

This paper continues the invèstigation of varieties of topological groups. It is shown that the family of all varieties of topological groups with any given underlying algebraic variety is a class and not a set. In fact the family of all β-varieties with any given underlying algebraic variety is a class and not a set. A variety generated by a family of topological groups of bounded cardinal is not a full variety.The varieties V(R) and V(T) generated by the additive group of reals and the circle group respectively each with its usual topology are examined. In particular it is shown that a locally compact Hausdorff abelian group is in V(T) if and only if it is compact. Thus V(R) properly contains V(T).It is proved that any free topological group of a non-indiscrete variety is disconnected. Finally, some comments are made on topologies on free groups.

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