Cellularity in quotient spaces of topological groups

2019 ◽  
Vol 31 (2) ◽  
pp. 351-359
Author(s):  
Mikhail G. Tkachenko
Keyword(s):  
Topological Group ◽  
Quotient Space ◽  
Compact Subgroup ◽  
Topological Groups ◽  
Quotient Spaces

AbstractLet G be an arbitrary topological group. We prove that the cellularity of G is equal to the cellularity of the quotient space {G/K} for every compact subgroup K of G.

scholarly journals Semi-quotient mappings and spaces

2016 ◽  
Vol 14 (1) ◽  
pp. 1014-1022 ◽  
Author(s):  
Moiz ud Din Khan ◽  
Rafaqat Noreen ◽  
Muhammad Siddique Bosan
Keyword(s):  
Quotient Space ◽  
Topological Groups ◽  
Quotient Spaces ◽  
Continuous Mappings ◽  
Quotient Mappings

AbstractIn this paper, we continue the study of s-topological and irresolute-topological groups. We define semi-quotient mappings which are stronger than semi-continuous mappings, and then consider semi-quotient spaces and groups. It is proved that for some classes of irresolute-topological groups (G, *, τ) the semi-quotient space G/H is regular. Semi-isomorphisms of s-topological groups are also discussed.

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.

Classification Algorithm of Case Retrieval Based on Granularity Calculation of Quotient Space

2020 ◽  
pp. 2150003
Author(s):  
Jiaxing Lu ◽  
Jiang Qing ◽  
Huang He ◽  
Zhang Zhengyong ◽  
Wang Rujing
Keyword(s):  
Quotient Space ◽  
Inner Product ◽  
Retrieval Algorithm ◽  
Case Based Reasoning ◽  
Case Retrieval ◽  
Current Case ◽  
Quotient Spaces ◽  
Similarity Calculation ◽  
Key Steps ◽  
Case Based

Case retrieval is one of the key steps of case-based reasoning. The quality of case retrieval determines the effectiveness of the system. The common similarity calculation methods based on attributes include distance and inner product. Different similarity calculations have different influences on the effect of case retrieval. How to combine different similarity calculation results to get a more widely used and better retrieval algorithm is a hot issue in the current case-based reasoning research. In this paper, the granularity of quotient space is introduced into the similarity calculation based on attribute, and a case retrieval algorithm based on granularity synthesis theory is proposed. This method first uses similarity calculation of different attributes to get different results of case retrieval, and considers that these classification results constitute different quotient spaces, and then organizes these quotient spaces according to granularity synthesis theory to get the classification results of case retrieval. The experimental results verify the validity and correctness of this method and the application potential of granularity calculation of quotient space in case-based reasoning.

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