Coarse Geometry of Topological Groups

2021 ◽  
Author(s):  
Christian Rosendal
Keyword(s):  
Geometric Structure ◽  
Metric Spaces ◽  
Automorphism Groups ◽  
Mathematical Practice ◽  
Worked Examples ◽  
Topological Groups ◽  
Diffeomorphism Groups ◽  
Locally Compact ◽  
Open Problems ◽  
Topological Transformation

This book provides a general framework for doing geometric group theory for many non-locally-compact topological transformation groups that arise in mathematical practice, including homeomorphism and diffeomorphism groups of manifolds, isometry groups of separable metric spaces and automorphism groups of countable structures. Using Roe's framework of coarse structures and spaces, the author defines a natural coarse geometric structure on all topological groups. This structure is accessible to investigation, especially in the case of Polish groups, and often has an explicit description, generalising well-known structures in familiar cases including finitely generated discrete groups, compactly generated locally compact groups and Banach spaces. In most cases, the coarse geometric structure is metrisable and may even be refined to a canonical quasimetric structure on the group. The book contains many worked examples and sufficient introductory material to be accessible to beginning graduate students. An appendix outlines several open problems in this young and rich theory.

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.

Graphical structure of extended b-metric spaces: an application to the transverse oscillations of a homogeneous bar

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Mudasir Younis ◽  
Deepak Singh ◽  
Ishak Altun ◽  
Varsha Chauhan
Keyword(s):  
Metric Spaces ◽  
Fixed Point Theory ◽  
Directed Graphs ◽  
Point Theory ◽  
Graph Structure ◽  
Open Ball ◽  
Open Problems ◽  
Graphical Structure ◽  
State Of Art ◽  
Transverse Oscillations

Abstract The purpose of this article is to present the notion of graphical extended b-metric spaces, blending the concepts of graph theory and metric fixed point theory. We discuss the structure of an open ball of the new proposed space and elaborate on the newly introduced ideas in a novel way by portraying suitably directed graphs. We also provide some examples in graph structure to show that our results are sharp as compared to the results in the existing state-of-art. Furthermore, an application to the transverse oscillations of a homogeneous bar is entrusted to affirm the applicability of the established results. Additionally, we evoke some open problems for enthusiastic readers for the future aspects of the study.

scholarly journals Density in locally compact topological groups

2014 ◽  
Vol 165 ◽  
pp. 26-38 ◽  
Author(s):  
Mohammad Akbari tootkaboni ◽  
Zeinab Eslami
Keyword(s):  
Topological Groups ◽  
Locally Compact ◽  
Locally Compact Topological Groups

Two open problems in the fixed point theory of contractive type mappings on quasimetric spaces

2017 ◽  
Vol 33 (2) ◽  
pp. 169-180
Author(s):  
MITROFAN M. CHOBAN ◽  
◽  
VASILE BERINDE ◽  
◽  
Keyword(s):  
Fixed Point ◽  
Metric Spaces ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Partial Answer ◽  
Open Problems ◽  
Complete Answer ◽  
Contractive Type ◽  
Generalized Distances

Two open problems in the fixed point theory of quasi metric spaces posed in [Berinde, V. and Choban, M. M., Generalized distances and their associate metrics. Impact on fixed point theory, Creat. Math. Inform., 22 (2013), No. 1, 23–32] are considered. We give a complete answer to the first problem, a partial answer to the second one, and also illustrate the complexity and relevance of these problems by means of four very interesting and comprehensive examples.

scholarly journals On Continuous Isomorphisms of Topological Groups

1950 ◽  
Vol 1 ◽  
pp. 109-111
Author(s):  
Morikuni Gotô ◽  
Hidehiko Yamabe
Keyword(s):  
Uniform Convergence ◽  
Topological Groups ◽  
Natural Topology ◽  
Locally Compact ◽  
Connected Group ◽  
Inner Automorphisms

Let G be a locally compact connected group, and let A (G) be the group of all continuous automorphisms of G. We shall introduce a natural topology into A(G) as previously (i.e. the topology of uniform convergence in the wider sense.) When the component of the identity of A(G) coincides with the group of inner automorphisms, we shall call G complete. The purpose of this note is to prove the following theorem and give some applications of it.

scholarly journals Relevant Arithmetic and Mathematical Pluralism

2021 ◽  
Vol 18 (5) ◽  
pp. 569-596
Author(s):  
Zach Weber
Keyword(s):  
Mathematical Practice ◽  
Worked Examples ◽  
Incompleteness Theorem ◽  
Classical Counterpart ◽  
The One ◽  
Mathematical Pluralism

In The Consistency of Arithmetic and elsewhere, Meyer claims to “repeal” Goedel’s second incompleteness theorem. In this paper, I review his argument, and then consider two ways of understanding it: from the perspective of mathematical pluralism and monism, respectively. Is relevant arithmetic just another legitimate practice among many, or is it a rival of its classical counterpart—a corrective to Goedel, setting us back on the path to the (One) True Arithmetic? To help answer, I sketch a few worked examples from relevant mathematics, to see what a non-classical (re)formulation of mathematics might look like in practice. I conclude that, while it is unlikely that relevant arithmetic describes past and present mathematical practice, and so might be most acceptable as a pluralist enterprise, it may yet prescribe a more monistic future venture.

Nowhere Dense Subsets of Metric Spaces with Applications to Stone-Cech Compactifications

1972 ◽  
Vol 24 (4) ◽  
pp. 622-630 ◽  
Author(s):  
Jack R. Porter ◽  
R. Grant Woods
Keyword(s):  
Metric Space ◽  
Hausdorff Space ◽  
Metric Spaces ◽  
Dense Subset ◽  
Locally Compact ◽  
Completely Regular ◽  
Isolated Points ◽  
Remote Points ◽  
Topological Boundary ◽  
Regular Closed

Let X be a metric space. Assume either that X is locally compact or that X has no more than countably many isolated points. It is proved that if F is a nowhere dense subset of X, then it is regularly nowhere dense (in the sense of Katětov) and hence is contained in the topological boundary of some regular-closed subset of X. This result is used to obtain new properties of the remote points of the Stone-Čech compactification of a metric space without isolated points.Let βX denote the Stone-Čech compactification of the completely regular Hausdorff space X. Fine and Gillman [3] define a point p of βX to be remote if p is not in the βX-closure of a discrete subset of X.

Simple Links in Locally Compact Connected Hausdorff Spaces are Nondegenerate

1981 ◽  
Vol 34 (2) ◽  
pp. 349-355
Author(s):  
David John
Keyword(s):  
Hausdorff Space ◽  
Metric Spaces ◽  
Connected Space ◽  
Hausdorff Spaces ◽  
Locally Compact ◽  
Peano Continua

The fact that simple links in locally compact connected metric spaces are nondegenerate was probably first established by C. Kuratowski and G. T. Whyburn in [2], where it is proved for Peano continua. J. L. Kelley in [3] established it for arbitrary metric continua, and A. D. Wallace extended the theorem to Hausdorff continua in [4]. In [1], B. Lehman proved this theorem for locally compact, locally connected Hausdorff spaces. We will show that the locally connected property is not necessary.A continuum is a compact connected Hausdorff space. For any two points a and b of a connected space M, E(a, b) denotes the set of all points of M which separate a from b in M. The interval ab of M is the set E(a, b) ∪ {a, b}.

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