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 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.

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 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 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.

scholarly journals On group uniformities on the square of a space and extending pseudometrics

1995 ◽  
Vol 51 (2) ◽  
pp. 309-335 ◽  
Author(s):  
Michael G. Tkačnko
Keyword(s):  
Topological Group ◽  
Topological Groups ◽  
Free Topological Group

We give some conditions under which, for a given pair (d1, d2) of continuous pseudometrics respectively on X and X3, there exists a continuous semi-norm N on the free topological group F(X) such that N(x · y−1) = d1(x, y) and N(x · y · t−1 · z−1) ≥ d2((x, y), (z, t)) for all x, y, z, t ∈ X. The “extension” results are applied to characterise thin subsets of free topological groups and obtain some relationships between natural uniformities on X2 and those induced by the group uniformities *V, V* and *V* of F(X).

scholarly journals VARIETIES OF ABELIAN TOPOLOGICAL GROUPS AND SCATTERED SPACES

2008 ◽  
Vol 78 (3) ◽  
pp. 487-495 ◽  
Author(s):  
CAROLYN E. MCPHAIL ◽  
SIDNEY A. MORRIS
Keyword(s):  
Topological Group ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Topological Groups ◽  
Abelian Topological Group ◽  
Free Abelian Topological Group ◽  
Scattered Spaces

AbstractThe variety of topological groups generated by the class of all abelian kω-groups has been shown to equal the variety of topological groups generated by the free abelian topological group on [0, 1]. In this paper it is proved that the free abelian topological group on a compact Hausdorff space X generates the same variety if and only if X is not scattered.

Open subgroups of free abelian topological groups

1993 ◽  
Vol 114 (3) ◽  
pp. 439-442 ◽  
Author(s):  
Sidney A. Morris ◽  
Vladimir G. Pestov
Keyword(s):  
Topological Group ◽  
Sufficient Conditions ◽  
Open Subgroup ◽  
Regular Space ◽  
Topological Groups ◽  
Covering Dimension ◽  
Abelian Topological Group ◽  
Completely Regular ◽  
Free Topological Group ◽  
Free Abelian Topological Group

We prove that any open subgroup of the free abelian topological group on a completely regular space is a free abelian topological group. Moreover, the free topological bases of both groups have the same covering dimension. The prehistory of this result is as follows. The celebrated Nielsen–Schreier theorem states that every subgroup of a free group is free, and it is equally well known that every subgroup of a free abelian group is free abelian. The analogous result is not true for free (abelian) topological groups [1,5]. However, there exist certain sufficient conditions for a subgroup of a free topological group to be topologically free [2]; in particular, an open subgroup of a free topological group on a kω-space is topologically free. The corresponding question for free abelian topological groups asked 8 years ago by Morris [11] proved to be more difficult and remained open even within the realm of kω-spaces. In the present paper a comprehensive answer to this question is obtained.

scholarly journals Free products of topological groups with a closed subgroup amalgamated

1986 ◽  
Vol 40 (3) ◽  
pp. 414-420 ◽  
Author(s):  
Peter Nickolas
Keyword(s):  
Topological Group ◽  
Free Product ◽  
Closed Subgroup ◽  
Countable Family ◽  
Topological Groups ◽  
Free Products ◽  
Dense Image ◽  
Amalgamated Free Product

AbstractIt is shown that if {Gn: n = 1, 2,…} is a countable family of Hausdorff kω-topological groups with a common closed subgroup A, then the topological amalgamated free product *AGn exists and is a Hausdorff kω-topological group with each Gn as a closed subgroup. A consequence is the theorem of La Martin that epimorphisms in the category of kω-topological groups have dense image.

Sign in / Sign up

Export Citation Format

Share Document