scholarly journals Topological groups with dense compactly generated subgroups

2002 ◽  
Vol 3 (1) ◽  
pp. 85 ◽  
Author(s):  
Hiroshi Fujita ◽  
Dimitri Shakhmatov
Keyword(s):  
Topological Group ◽  
Compact Subset ◽  
Open Subset ◽  
Identity Element ◽  
Topological Groups ◽  
Finitely Generated ◽  
Finite Set ◽  
Metrizable Topological Group ◽  
Compactly Generated ◽  
Compact Subsets

<p>A topological group G is: (i) compactly generated if it contains a compact subset algebraically generating G, (ii) -compact if G is a union of countably many compact subsets, (iii) <sub>0</sub>-bounded if arbitrary neighborhood U of the identity element of G has countably many translates xU that cover G, and (iv) finitely generated modulo open sets if for every non-empty open subset U of G there exists a finite set F such that F  U algebraically generates G. We prove that: (1) a topological group containing a dense compactly generated subgroup is both <sub>0</sub>-bounded and finitely generated modulo open sets, (2) an almost metrizable topological group has a dense compactly generated subgroup if and only if it is both <sub>0</sub>-bounded and finitely generated modulo open sets, and (3) an almost metrizable topological group is compactly generated if and only if it is -compact and finitely generated modulo open sets.</p>

Finite presentability of generalized Bruck–Reilly ∗-extension of groups

2016 ◽  
Vol 09 (04) ◽  
pp. 1650090 ◽  
Author(s):  
Seda Oğuz ◽  
Eylem G. Karpuz
Keyword(s):  
Finite Subset ◽  
Identity Element ◽  
Finitely Generated ◽  
Finite Generation ◽  
Finite Set ◽  
Finite Presentability ◽  
Finitely Presented

In [Finite presentability of Bruck–Reilly extensions of groups, J. Algebra 242 (2001) 20–30], Araujo and Ruškuc studied finite generation and finite presentability of Bruck–Reilly extension of a group. In this paper, we aim to generalize some results given in that paper to generalized Bruck–Reilly ∗-extension of a group. In this way, we determine necessary and sufficent conditions for generalized Bruck–Reilly ∗-extension of a group, [Formula: see text], to be finitely generated and finitely presented. Let [Formula: see text] be a group, [Formula: see text] be morphisms and [Formula: see text] ([Formula: see text] and [Formula: see text] are the [Formula: see text]- and [Formula: see text]-classes, respectively, contains the identity element [Formula: see text] of [Formula: see text]). We prove that [Formula: see text] is finitely generated if and only if there exists a finite subset [Formula: see text] such that [Formula: see text] is generated by [Formula: see text]. We also prove that [Formula: see text] is finitely presented if and only if [Formula: see text] is presented by [Formula: see text], where [Formula: see text] is a finite set and [Formula: see text] [Formula: see text] for some finite set of relations [Formula: see text].

scholarly journals UNIVERSAL SUBGROUPS OF POLISH GROUPS

2014 ◽  
Vol 79 (4) ◽  
pp. 1148-1183 ◽  
Author(s):  
KONSTANTINOS A. BEROS
Keyword(s):  
Topological Group ◽  
Banach Spaces ◽  
Topological Groups ◽  
Locally Compact ◽  
Polish Groups ◽  
Countable Power ◽  
Compact Case ◽  
Image Position ◽  
The Relationship ◽  
Compactly Generated

AbstractGiven a class${\cal C}$of subgroups of a topological groupG, we say that a subgroup$H \in {\cal C}$is auniversal${\cal C}$subgroupofGif every subgroup$K \in {\cal C}$is a continuous homomorphic preimage ofH. Such subgroups may be regarded as complete members of${\cal C}$with respect to a natural preorder on the set of subgroups ofG. We show that for any locally compact Polish groupG, the countable powerGωhas a universalKσsubgroup and a universal compactly generated subgroup. We prove a weaker version of this in the nonlocally compact case and provide an example showing that this result cannot readily be improved. Additionally, we show that many standard Banach spaces (viewed as additive topological groups) have universalKσand compactly generated subgroups. As an aside, we explore the relationship between the classes ofKσand compactly generated subgroups and give conditions under which the two coincide.

Weakly compactly generated locally convex spaces

1977 ◽  
Vol 82 (1) ◽  
pp. 85-98 ◽  
Author(s):  
Richard J. Hunter ◽  
John Lloyd
Keyword(s):  
Banach Spaces ◽  
Compact Subset ◽  
Locally Convex Spaces ◽  
Weakly Compact ◽  
Permanence Properties ◽  
Locally Convex ◽  
Compactly Generated ◽  
Convex Spaces ◽  
Compact Subsets

AbstractLocally convex spaces which are generated by a weakly compact subset (or a sequence of weakly compact subsets) and their subspaces are studied. Various characterizations and the permanence properties of these spaces are obtained. Certain results valid for weakly compactly generated Banach spaces are extended. These spaces are shown to have sequential properties which extend well-known properties of separable locally convex spaces.

scholarly journals Pseudobounded or ω-pseudobounded paratopological groups

Filomat ◽  
2011 ◽  
Vol 25 (3) ◽  
pp. 93-103 ◽  
Author(s):  
Fucai Lin ◽  
Shou Lin
Keyword(s):  
Topological Group ◽  
Natural Number ◽  
Identity Element ◽  
Topological Groups ◽  
Open Problems ◽  
Paratopological Group

We say that a paratopological group G is pseudobounded (?-pseudobounded), if for every neighborhood V of the identity element e of G, there exists a natural number n such that G=Vn (G = U?n=1 Vn). In this paper, we mainly discuss the pseudobounded and ?-pseudobounded paratopological groups. First, we give an example to show that a theorem in [4] is not true. And then, we define the concept of premeager, and discuss when a pseudobounded paratopological group is a topological group. Moreover, we also discuss some properties of ?-pseudobounded topological groups, and show that the class of connected topological groups is contained in the class of ?-pseudobounded topological groups. Finally, some open problems concerning the paratopological groups are posed.

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.

Neutrosophic Bi-topological Group

2021 ◽  
pp. 61-67
Author(s):  
Riad K. Al Al-Hamido ◽  
Keyword(s):  
Topological Group ◽  
Continuous Group ◽  
Topological Groups ◽  
Basic Properties ◽  
New Models

Neutrosophic topological groups are neutrosophic groups in an algebraic sense together with neutrosophic continuous group operations. In this article, we have presented neutrosophic bi-topological groups with illustrative examples. We have also defined eight new models of neutrosophic bi-topological groups. Neutrosophic bi-topological group that depends on two neutrosophic topologies group is more general than the neutrosophic topological group. Finally, Some basic properties of neutrosophic bi-topological groups were studied.

scholarly journals Compact Subsets of the Glimm Space of a C*-algebra

2014 ◽  
Vol 57 (1) ◽  
pp. 90-96
Author(s):  
Aldo J. Lazar
Keyword(s):  
Compact Subset ◽  
Positive Element ◽  
C Algebra ◽  
Compact Subsets

AbstractIf A is a σ-unital C*-algebra and a is a strictly positive element of A, then for every compact subset K of the complete regularization Glimm(A) of Prim(A) there exists α > 0 such that K ⊂ {G ∊ Glimm(A) | ||a + G|| ≥ α: This extends a result of J. Dauns to all σ-unital C*-algebras. However, there exist a C*-algebra A and a compact subset of Glimm(A) that is not contained in any set of the form {G ∊ Glimm(A) | ||a + G|| ≥}, a ∊ A and α > 0.

ON THE RANK OF ABELIAN VARIETIES OVER AMPLE FIELDS

2010 ◽  
Vol 06 (03) ◽  
pp. 579-586 ◽  
Author(s):  
ARNO FEHM ◽  
SEBASTIAN PETERSEN
Keyword(s):  
Finite Field ◽  
Abelian Variety ◽  
Galois Extension ◽  
Characteristic Zero ◽  
Rational Point ◽  
Abelian Varieties ◽  
Finitely Generated ◽  
Infinite Rank ◽  
Finite Set

A field K is called ample if every smooth K-curve that has a K-rational point has infinitely many of them. We prove two theorems to support the following conjecture, which is inspired by classical infinite rank results: Every non-zero Abelian variety A over an ample field K which is not algebraic over a finite field has infinite rank. First, the ℤ(p)-module A(K) ⊗ ℤ(p) is not finitely generated, where p is the characteristic of K. In particular, the conjecture holds for fields of characteristic zero. Second, if K is an infinite finitely generated field and S is a finite set of local primes of K, then every Abelian variety over K acquires infinite rank over certain subfields of the maximal totally S-adic Galois extension of K. This strengthens a recent infinite rank result of Geyer and Jarden.

scholarly journals More on generalized topological groups

2013 ◽  
Vol 22 (1) ◽  
pp. 47-51
Author(s):  
MURAD HUSSAIN ◽  
◽  
MOIZ UD DIN KHAN ◽  
CENAP OZEL ◽  
◽  
...  
Keyword(s):  
Topological Group ◽  
Group Structure ◽  
Topological Groups

In the paper [Hussain, M., Khan, M. and Ozel, C., ¨ On Generalized Topological Groups] we defined the generalized topological group structure and we proved some basic results. In this work we introduce the notions of ultra Hausdorffness and ultra G-Hausdorffness and we give the relation between the ultra G-Hausdorffness and G-compactness.

ON THE RELATIONSHIPS BETWEEN VARIOUS LATTICE-VALUED TOPOLOGICAL GROUPS AND THEIR UNIFORMITIES

2012 ◽  
Vol 08 (03) ◽  
pp. 361-383
Author(s):  
J. AL-MUFARRIJ ◽  
T. M. G. AHSANULLAH
Keyword(s):  
Topological Group ◽  
Topological Groups ◽  
The Relationship

The purpose of this article is to investigate the relationships between some of the lattice-valued topological groups, and the lattice-valued uniformities that they inherit. In so doing, we look at the relationship between (a) crisp sets of lattice-valued neighborhood groups and lattice-valued neighborhood topological groups, and their uniformities; (b) lattice-valued topological groups of ordinary subsets and fuzzy neighborhood groups, and their uniformities. We also investigate the connection between stratified lattice-valued neighborhood topological group and its level spaces.

Sign in / Sign up

Export Citation Format

Share Document