• Search Research Projects
  • Search Researchers
  • How to Use
  1. Back to previous page

Study of the structure of the Markov-Zariski topology of a group and convergence properties of compact-like topological groups

Research Project

Project/Area Number 19540092
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Geometry
Research InstitutionEhime University

Principal Investigator

SHAKHMATOV Dmitri B.  Ehime University, 理工学研究科, 教授 (90253294)

Co-Investigator(Kenkyū-buntansha) NOGURA Tsugunori  愛媛大学, 理工学研究科, 教授 (00036419)
FUJITA Hiroshi  愛媛大学, 理工学研究科, 講師 (60238582)
Project Period (FY) 2007 – 2009
Project Status Completed (Fiscal Year 2009)
Budget Amount *help
¥4,290,000 (Direct Cost: ¥3,300,000、Indirect Cost: ¥990,000)
Fiscal Year 2009: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Fiscal Year 2008: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
Fiscal Year 2007: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
Keywordsトポロジー / 代数学 / 位相群 / コンパクト / 収束性
Research Abstract

We prove that Markov and Zariski topologies coincide for abelian groups, and we provide the description of the Markov-Zariski topology of an abelian group. Based on this description, we characterize counatable potentially dense susbets of abelian groups of size at most the continuum, as well as uncountable potentially dense subsets of torsion and divisible abelian groups. We also find a necessary and sufficient condition for the coincidence of Markov and Zariski topologies in the non-commutative case. The theory of group-valued function spaces is developed.

Report

(4 results)
  • 2009 Annual Research Report   Final Research Report ( PDF )
  • 2008 Annual Research Report
  • 2007 Annual Research Report
  • Research Products

    (49 results)

All 2010 2009 2008 2007 Other

All Journal Article (31 results) (of which Peer Reviewed: 29 results) Presentation (14 results) Remarks (4 results)

  • [Journal Article] Hewitt-Marczewski-Pondiczery type theorem for abelian groups and Markov's potential density2010

    • Author(s)
      D. Dikranjan; D. Shakhmatov
    • Journal Title

      Proceedings of the American Mathematical Society 138

      Pages: 2979-2990

    • Related Report
      2009 Final Research Report
    • Peer Reviewed
  • [Journal Article] Group-valued continuous functions with the topology of pointwise convergence2010

    • Author(s)
      D. Shakhmatov; J. Spevak
    • Journal Title

      Topology and its Applications 157

      Pages: 1518-1540

    • Related Report
      2009 Final Research Report
    • Peer Reviewed
  • [Journal Article] Quasiconvex density and determining subgroups of compact abelian groups (with D. Dikranjan)2010

    • Author(s)
      D. Dikranjan; D. Shakhmatov
    • Journal Title

      Journal of Mathematical Analysis and Applications 363

      Pages: 42-48

    • Related Report
      2009 Final Research Report
    • Peer Reviewed
  • [Journal Article] Weak orderability of topological spaces2010

    • Author(s)
      V. Gutev; T. Nogura
    • Journal Title

      Topology and its Applications 157

      Pages: 1249-1274

    • Related Report
      2009 Final Research Report
    • Peer Reviewed
  • [Journal Article] Set-maximal selections2010

    • Author(s)
      V. Gutev; T. Nogura
    • Journal Title

      Topology and its Applications 157

      Pages: 53-61

    • Related Report
      2009 Final Research Report
    • Peer Reviewed
  • [Journal Article] On difference property of Borel measurable functions2010

    • Author(s)
      H. Fujita; T. Matrai
    • Journal Title

      Fundamenta Mathematicae 208

      Pages: 57-73

    • Related Report
      2009 Final Research Report
    • Peer Reviewed
  • [Journal Article] Quasi-convex density and determining subgroups of compact abelian groups2010

    • Author(s)
      D.Dikranjan, D.Shakhmatov
    • Journal Title

      Journal of Mathematical Analysis and its Applications 363

      Pages: 42-48

    • Related Report
      2009 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Group-valued continuous functions with the topology of pointwise convergence2010

    • Author(s)
      D.Shakhmatov, J.Spevak
    • Journal Title

      Topology and its Applications 157

      Pages: 1518-1540

    • Related Report
      2009 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Hewitt-Marczewski-Pondiczery type theorem for abelian groups and Markov's potential density2010

    • Author(s)
      D.Dikranjan, D.Shakhmatov
    • Journal Title

      Proceedings of the American Mathematical Society 138

      Pages: 2979-2990

    • Related Report
      2009 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Minimal pseudocompact group topologies on free abelian groups2009

    • Author(s)
      D. Dikranjan; A. Giordano Bruno; D. Shakhmatov
    • Journal Title

      Topology and its Applications 156

      Pages: 2039-2053

    • Related Report
      2009 Final Research Report
    • Peer Reviewed
  • [Journal Article] Building suitable sets for locally compact groups by means of continuous selections2009

    • Author(s)
      D. Shakhmatov
    • Journal Title

      Topology and its Applications 156

      Pages: 1216-1223

    • Related Report
      2009 Final Research Report
    • Peer Reviewed
  • [Journal Article] Weak regularity and consecutive topologizations and regularizations of pretopologies2009

    • Author(s)
      S. Dolecki; H. Kunzi; T. Nogura
    • Journal Title

      Topology and its Applications 156

      Pages: 1306-1314

    • Related Report
      2009 Final Research Report
    • Peer Reviewed
  • [Journal Article] Minimal pseudocompact group topologies on free abelian groups2009

    • Author(s)
      D.Dikranjan, A.Giordano Bruno, D.Shakhmatov
    • Journal Title

      Topology and its Applications 156

      Pages: 2039-2053

    • Related Report
      2009 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Building suitable sets for locally compact groups by Means of continuous selections2009

    • Author(s)
      D. Shakhmatov
    • Journal Title

      Topology and its Applications 156

      Pages: 1216-1223

    • Related Report
      2008 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Weak regularity and consecutive topologizations and regularizations of pretopologies2009

    • Author(s)
      S. Dolecki, H. Kunzi, T. Nogura
    • Journal Title

      Topology and its Applications 156

      Pages: 1306-1314

    • Related Report
      2008 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Reflection principle characterizing groups in which unconditionally closed sets are algebraic2008

    • Author(s)
      D. Dikranjan; D. Shakhmatov
    • Journal Title

      Journal of Group Theory 11,no.3

      Pages: 421-442

    • Related Report
      2009 Final Research Report
    • Peer Reviewed
  • [Journal Article] Abundance of sigma-compact non-compactly generated groups witnessed by the Bohr topology of an abelian group2008

    • Author(s)
      D. Dikranjan; D. Shakhmatov
    • Journal Title

      数理解析研究所講究緑, 一般及び幾何学的トポロジーの現状と諸問題 1578

      Pages: 16-18

    • Related Report
      2009 Final Research Report
  • [Journal Article] Extensions of 2-point selections2008

    • Author(s)
      S. Garcia-Ferreira; V. Gutev; T. Nogura
    • Journal Title

      New Zealand Journal of Mathematics 38

      Pages: 1-8

    • Related Report
      2009 Final Research Report
    • Peer Reviewed
  • [Journal Article] Selections and totally disconnected spaces2008

    • Author(s)
      V. Gutev; T. Nogura
    • Journal Title

      Topology and its Applications 155

      Pages: 824-829

    • Related Report
      2009 Final Research Report
    • Peer Reviewed
  • [Journal Article] Weak selections and flows in networks2008

    • Author(s)
      V. Gutev; T. Nogura
    • Journal Title

      Commentationes Mathematicae Universitatis Carolinae 49

      Pages: 284-289

    • Related Report
      2009 Final Research Report
    • Peer Reviewed
  • [Journal Article] Reflection principle characterizing groups in which all unconditionally closed sets are algebraic2008

    • Author(s)
      D. Dikranjan, D, Shakhmatov
    • Journal Title

      Journal of Group Theory 11

      Pages: 421-442

    • Related Report
      2008 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Selections and totally disconnected spaces2008

    • Author(s)
      V. Gutev, T. Nogura
    • Journal Title

      Topology and its Applications 155

      Pages: 824-829

    • Related Report
      2008 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Weak selections and flows in networks2008

    • Author(s)
      V. Gutev, T. Nogura
    • Journal Title

      Commentationes Mathematicae Universitatis Carolinae 49

      Pages: 824-829

    • Related Report
      2008 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Extensions of 2-point selections2008

    • Author(s)
      V. Gutev, S. Garcia-Ferreira, T. Nogura
    • Journal Title

      New Zealand Journal of Mathematics 38

      Pages: 1-8

    • Related Report
      2008 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Abundance of sigma-compact non-compactly generated groups witnessed by the Bohr topology of an abelian group2008

    • Author(s)
      D. Dikranjan, D. Shakhmatov
    • Journal Title

      数理解析研究所講究緑, 一般及び幾何学的トポロジーの現状と諸問題 1578

      Pages: 16-18

    • Related Report
      2007 Annual Research Report
  • [Journal Article] Remarks on two problems by M. Laczkovich on functions with Borel measurable differences2007

    • Author(s)
      H. Fujita
    • Journal Title

      Acta Mathematica Hungarica 117

      Pages: 153-160

    • Related Report
      2009 Final Research Report
    • Peer Reviewed
  • [Journal Article] Convergence properties of hyperspaces2007

    • Author(s)
      G. Di Maio; Lj Kocinac; T. Nogura
    • Journal Title

      Journal of the Korean Mathematical Society 44

      Pages: 845-854

    • Related Report
      2009 Final Research Report
    • Peer Reviewed
  • [Journal Article] Remarks on two problems by M.Laczkovich on functions with Borel measurable differences2007

    • Author(s)
      H. Fujita
    • Journal Title

      Acta Mathematica Hungarica 117

      Pages: 153-160

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Convergence properties of hyperspaces2007

    • Author(s)
      G. di Maio, Lj. Kocinac, T. Nogura
    • Journal Title

      Journal of Korean Mathematical Society 44

      Pages: 845-854

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Journal Article] The Markov-Zariski topology of an abelian group

    • Author(s)
      D. Dikranjan; D. Shakhmatov
    • Journal Title

      Journal of Algebra (印刷中)

    • Related Report
      2009 Final Research Report
    • Peer Reviewed
  • [Journal Article] The Markov-Zariski topology of an abelian group

    • Author(s)
      D.Dikranjan, D.Shakhmatov
    • Journal Title

      Journal of Algebra (印刷中)

    • Related Report
      2009 Annual Research Report
    • Peer Reviewed
  • [Presentation] Hewitt-Marczewski- Pondiczery type theorem for abelian groups and Markov's potential density2009

    • Author(s)
      D. Shakhmatov
    • Organizer
      The 24th Summer Conference on Topology and its Applications
    • Place of Presentation
      Brno, Czech Republic
    • Year and Date
      2009-07-17
    • Related Report
      2009 Final Research Report
  • [Presentation] Hewitt-Marczewski-Pondiczery type theorem for abelian groups and Markov's potential density2009

    • Author(s)
      D.Shakhmatov
    • Organizer
      The 24^<th> Summer Conference on Topology and its Applicatoons
    • Place of Presentation
      Brno, Czech Republic
    • Year and Date
      2009-07-17
    • Related Report
      2009 Annual Research Report
  • [Presentation] Group-valued continuous functions with the topology of point-wise convergence2009

    • Author(s)
      D. Shakhmatov; J. Spevak
    • Organizer
      大44回位相空間論シンポジウム
    • Place of Presentation
      島根大学
    • Year and Date
      2009-05-31
    • Related Report
      2009 Final Research Report
  • [Presentation] Selections and Dimension2009

    • Author(s)
      T. Nogura
    • Organizer
      大44回位相空間論シンポジウム
    • Place of Presentation
      島根大学
    • Year and Date
      2009-05-30
    • Related Report
      2009 Final Research Report
  • [Presentation] Advances in Set-Theoretic Topology2008

    • Author(s)
      D. Shakhmatov, Strong S, L groups
    • Organizer
      Centre "Ettore Majoranana"
    • Place of Presentation
      Centre "Ettore Majoranana", Erice, Italy
    • Year and Date
      2008-06-18
    • Related Report
      2009 Final Research Report
  • [Presentation] A partial answer to a problem of M. Laczkowich concerning the difference property of Borel measurable functions2008

    • Author(s)
      H. Fujita
    • Organizer
      Advances in Set-Theoretic Topology
    • Place of Presentation
      Centre "Ettore Majoranana", Erice, Italy
    • Year and Date
      2008-06-18
    • Related Report
      2009 Final Research Report
  • [Presentation] Strong S and L groups2008

    • Author(s)
      D. Shakhmatov
    • Organizer
      Advances in Set-theoretic Topology
    • Place of Presentation
      Cenre“Ettore Majoranana", Erice, Italy
    • Year and Date
      2008-06-18
    • Related Report
      2008 Annual Research Report
  • [Presentation] A partial answer to a problem of M. Laczkowich concerning the difference property of Borel measurable functions2008

    • Author(s)
      H. Fuiita
    • Organizer
      Advances in Set-theoretic Topology
    • Place of Presentation
      Centre“Ettore Majoranana", Erice, Italy
    • Year and Date
      2008-06-18
    • Related Report
      2008 Annual Research Report
  • [Presentation] Extreme selections and dimensions2008

    • Author(s)
      T. Nogura
    • Organizer
      Advances in Set-theoretic Topology
    • Place of Presentation
      Centre“Ettore Majoranana", Erice, Italy
    • Year and Date
      2008-06-10
    • Related Report
      2008 Annual Research Report
  • [Presentation] Continuous selections and Dimensions2008

    • Author(s)
      T. Nogura
    • Organizer
      Advances in Set-Theoretic Topology
    • Place of Presentation
      Centre "Ettore Majoranana", Erice, Italy(招待講演)
    • Year and Date
      2008-06-09
    • Related Report
      2009 Final Research Report
  • [Presentation] Solutions to 1944 problems of Markov2007

    • Author(s)
      D. Shakhmatov
    • Organizer
      International Conference on Topology and its Applications 2007
    • Place of Presentation
      京都大学(招待講演)
    • Year and Date
      2007-12-05
    • Related Report
      2009 Final Research Report
  • [Presentation] Selections and Dimensions2007

    • Author(s)
      T. Nogura
    • Organizer
      22th Summer Conference on Topology and its Applications
    • Place of Presentation
      Castellon, Spain
    • Year and Date
      2007-07-25
    • Related Report
      2009 Final Research Report
  • [Presentation] Solutions to 1944 problems of Markov2007

    • Author(s)
      D.Shakhmatov
    • Organizer
      International Conference on Topology and its Applications 2007
    • Place of Presentation
      京都大学
    • Related Report
      2007 Annual Research Report
  • [Presentation] Selections and dimensions2007

    • Author(s)
      野倉 嗣紀
    • Organizer
      22nd Summer Conference on Topology and its Applications
    • Place of Presentation
      Universidad Jaume I de Castellon,Spain
    • Related Report
      2007 Annual Research Report
  • [Remarks] 本研究の成果を以下のホームページで公開さらている

    • URL

      http://www.math.sci.ehime-u.ac.jp/~dima/

    • Related Report
      2009 Final Research Report
  • [Remarks]

    • URL

      http://www.math.sci.ehime-u.ac.jp/~dima/

    • Related Report
      2009 Annual Research Report
  • [Remarks]

    • URL

      http://www.math.sci.ehime-u.ac.jp~dima/

    • Related Report
      2008 Annual Research Report
  • [Remarks]

    • URL

      http://www.math.sci.ehime-u.ac.jp/~dima/

    • Related Report
      2007 Annual Research Report

URL: 

Published: 2007-04-01   Modified: 2021-04-07  

Information User Guide FAQ News Terms of Use Attribution of KAKENHI

Powered by NII kakenhi