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.

scholarly journals A study of some aspects of topological groups

Filomat ◽  
2007 ◽  
Vol 21 (1) ◽  
pp. 55-65
Author(s):  
M.R. Adhikari ◽  
M. Rahaman
Keyword(s):  
Topological Group ◽  
Group Structure ◽  
Base Point ◽  
Homotopy Category ◽  
Topological Spaces ◽  
Topological Groups ◽  
Homotopy Classes ◽  
Continuous Maps

The aim of this paper is to find a generalization of topological groups. The concept arises out of the investigation to obtain a group structure on the set [X,Y], of homotopy classes of maps from a space X to a given space Y for all X which is natural with respect to X. We also study the generalized topological groups. Finally, associated with each generalized topological group we construct a contra variant functor from the homotopy category of pointed topological spaces and base point preserving continuous maps to the category of groups and homomorphism.

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 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.

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.

scholarly journals A Schreier theorem for free topological groups

1975 ◽  
Vol 13 (1) ◽  
pp. 121-127 ◽  
Author(s):  
Peter Nickolas
Keyword(s):  
Topological Group ◽  
Closed Subgroup ◽  
Open Subgroup ◽  
Necessary Condition ◽  
Topological Groups ◽  
Free Topological Group

M.I.Graev has shown that subgroups of free topological groups need not be free. Brown and Hardy, however, have proved that any open subgroup of the free topological group on a kw-space is again a free topological group: indeed, this is true for any closed subgroup for which a Schreier transversal can be chosen continuously. This note provides a proof of this result more direct than that of Brown and Hardy. An example is also given to show that the condition stated in the theorem is not a necessary condition for freeness of a subgroup. Finally, a sharpened version of a particular case of the theorem is obtained, and is applied to the preceding example.

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