scholarly journals On generalized topological groups

Filomat ◽  
2013 ◽  
Vol 27 (4) ◽  
pp. 567-575 ◽  
Author(s):  
Murad Hussain ◽  
ud Khan ◽  
Cenap Özel
Keyword(s):  
Topological Group ◽  
Topological Space ◽  
Algebraic Structure ◽  
Topological Structure ◽  
Topological Groups ◽  
Generalized Topological Space ◽  
And Inversion

In this paper, we initiate the study of generalized topological groups. A generalized topological group has the algebraic structure of groups and the topological structure of a generalized topological space defined by A. Cs?sz?r [2] and they are joined together by the requirement that multiplication and inversion are G-continuous. Every topological group is a G-topological group whereas converse is not true in general. Quotients of generalized topological groups are defined and studied.

Varieties of topological groups

1969 ◽  
Vol 1 (2) ◽  
pp. 145-160 ◽  
Author(s):  
Sidney A. Morris
Keyword(s):  
Topological Group ◽  
Topological Space ◽  
Free Group ◽  
Quotient Group ◽  
Sufficient Conditions ◽  
Algebraic Theory ◽  
Topological Groups ◽  
Free Basis ◽  
Free Topological Group ◽  
Necessary And Sufficient

We introduce the concept of a variety of topological groups and of a free topological group F(X, ) of on a topological space X as generalizations of the analogous concepts in the theory of varieties of groups. Necessary and sufficient conditions for F(X, ) to exist are given and uniqueness is proved. We say the topological group FM,(X) is moderately free on X if its topology is maximal and it is algebraically free with X as a free basis. We show that FM(X) is a free topological group of the variety it generates and that if FM(X) is in then it is topologically isomorphic to a quotient group of F(X, ). It is also shown how well known results on free (free abelian) topological groups can be deduced. In the algebraic theory there are various equivalents of a free group of a variety. We examine the relationships between the topological analogues of these. In the appendix a result similar to the Stone-Čech compactification is proved.

scholarly journals A characterization of point semiuniformities

1996 ◽  
Vol 19 (2) ◽  
pp. 311-316
Author(s):  
Jennifer P. Montgomery
Keyword(s):  
Topological Group ◽  
Topological Space ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Topological Groups ◽  
Necessary And Sufficient

The concept of a uniformity was developed by A. Well and there have been several generalizations. This paper defines a point semiuniformity and gives necessary and sufficient conditions for a topological space to be point semiuniformizable. In addition, just as uniformities are associated with topological groups, a point semiuniformity is naturally associated with a semicontinuous group. This paper shows that a point semiuniformity associated with a semicontinuous group is a uniformity if and only if the group is a topological group.

scholarly journals Varieties of topological groups and left adjoint functors

1973 ◽  
Vol 16 (2) ◽  
pp. 220-227 ◽  
Author(s):  
Sidney A. Morris
Keyword(s):  
Topological Group ◽  
Topological Space ◽  
Left Adjoint ◽  
Topological Groups ◽  
Adjoint Functors ◽  
Free Topological Group

In [6] and [2] Markov and Graev introduced their respective concepts of a free topological group. Graev's concept is more general in the sense that every Markov free topological group is a Graev free topological group. In fact, if FG(X) is the Graev free topological group on a topological space X, then it is the Markov free topological group FM(Y) on some space Y if and only if X is disconnected. This, however, does not say how FG(X) and FM(X) are related.

TOPOLOGICAL GROUPS: VIRTUE OF PRE-OPEN SETS

2021 ◽  
Vol 10 (1) ◽  
pp. 385-389
Author(s):  
P. Gnanachandra ◽  
A.M Kumar
Keyword(s):  
Topological Group ◽  
Topological Groups ◽  
And Inversion

In this paper, we introduce notions of $\mathit{p}$-topological group and $\mathit{p}$-irresolute topological group which are generalizations of the notion topological group. We discuss the properties of $\mathit{p}$-topological group with illustrated examples. Also, we prove that translation and inversion in $\mathit{p}$-topological group are $\mathit{p}$-homeomorphism.

scholarly journals Locally dyadic topological groups

1989 ◽  
Vol 40 (3) ◽  
pp. 417-419 ◽  
Author(s):  
Joan Cleary ◽  
Sidney A. Morris
Keyword(s):  
Topological Group ◽  
Topological Space ◽  
Topological Groups ◽  
Locally Compact

A topological space is said to be locally dyadic if every neighbourhood of a point contains a dyadic neighbourhood of that point. It is proved here that every locally compact Hausdorff topological group is locally dyadic.

scholarly journals Semigroup structures for families of functions, I. Some Homomorphism Theorems

1967 ◽  
Vol 7 (1) ◽  
pp. 81-94 ◽  
Author(s):  
Kenneth D. Magill
Keyword(s):  
Continuous Function ◽  
Topological Space ◽  
Algebraic Structure ◽  
Topological Structure ◽  
Continuous Functions ◽  
The Family ◽  
Image Position ◽  
Natural Way

This is the first of several papers which grew out of an attempt to provide C (X, Y), the family of all continuous functions mapping a topological space X into a topological space Y, with an algebraic structure. In the event Y has an algebraic structure with which the topological structure is compatible, pointwise operations can be defined on C (X, Y). Indeed, this has been done and has proved extremely fruitful, especially in the case of the ring C (X, R) of all continuous, real-valued functions defined on X [3]. Now, one can provide C(X, Y) with an algebraic structure even in the absence of an algebraic structure on Y. In fact, each continuous function from Y into X determines, in a natural way, a semigroup structure for C(X, Y). To see this, let ƒ be any continuous function from Y into X and for ƒ and g in C(X, Y), define ƒg by each x in X.

STRATIFIED LATTICE-VALUED BALANCED NEIGHBORHOOD TOPOLOGICAL GROUPS

2012 ◽  
Vol 08 (02) ◽  
pp. 153-166
Author(s):  
JAWAHER AL-MUFARRIJ
Keyword(s):  
Topological Group ◽  
Topological Space ◽  
Uniform Space ◽  
Uniform Continuity ◽  
Topological Groups ◽  
Group Operation ◽  
Binary Group ◽  
Uniform Equicontinuity

The aim of this article is to introduce the notion of stratified lattice-valued balanced neighborhood topological group. We also introduce the notions of equicontinuity in stratified lattice-valued neighborhood topological space, and uniform equicontinuity in stratified lattice-valued uniform space. We use these notions to characterize stratified lattice-valued neighborhood topological groups. Moreover, introducing the notions of lattice-valued neighborhood open function and lattice-valued uniformly open function, we show that in a stratified lattice-valued neighborhood topological group these notions are equivalent. Finally, we conclude with a characterization of balanced stratified lattice-valued neighborhood topological group in terms of uniform continuity of binary group operation.

TOPOLOGICAL GROUPS: VIRTUE OF PRE-OPEN SETS

2021 ◽  
Vol 10 (1) ◽  
pp. 385-389
Author(s):  
P. Gnanachandra ◽  
A.M Kumar
Keyword(s):  
Topological Group ◽  
Topological Groups ◽  
And Inversion

In this paper, we introduce notions of $\mathit{p}$-topological group and $\mathit{p}$-irresolute topological group which are generalizations of the notion topological group. We discuss the properties of $\mathit{p}$-topological group with illustrated examples. Also, we prove that translation and inversion in $\mathit{p}$-topological group are $\mathit{p}$-homeomorphism.

scholarly journals An operation on topological spaces

2000 ◽  
Vol 1 (1) ◽  
pp. 13
Author(s):  
A.V. Arhangelskii
Keyword(s):  
Topological Space ◽  
Topological Structure ◽  
Necessary Conditions ◽  
Topological Spaces ◽  
Topological Groups ◽  
Compact Spaces ◽  
Product Operation ◽  
Made In

<p>A (binary) product operation on a topological space X is considered. The only restrictions are that some element e of X is a left and a right identity with respect to this multiplication, and that certain natural continuity requirements are satisfied. The operation is called diagonalization (of X). Two problems are considered: 1. When a topological space X admits such an operation, that is, when X is diagonalizable? 2. What are necessary conditions for diagonalizablity of a space (at a given point)? A progress is made in the article on both questions. In particular, it is shown that certain deep results about the topological structure of compact topological groups can be extended to diagonalizable compact spaces. The notion of a Moscow space is instrumental in our study.</p>

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.

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