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

Research of topological and geometric structures of topological groups

Research Project

Project/Area Number 12640065
Research Category

Grant-in-Aid for Scientific Research (C)

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

Principal Investigator

YAMADA Kohzo  Shizuoka University, Faculty of Education, Department of Mathematics, Assistant Professor, 教育学部, 助教授 (00200717)

Co-Investigator(Kenkyū-buntansha) SHAKHMATOV Dmitri  Ehime University, Faculty of Science, Department of Mathematics, Professor, 理学部, 教授 (90253294)
MIYATA Yoshimasa  Shizuoka University, Faculty of Education, Department of Mathematics, Professor, 教育学部, 教授 (50022207)
OHTA Haruto  Shizuoka University, Faculty of Education, Department of Mathematics, Professor, 教育学部, 教授 (40126769)
Project Period (FY) 2000 – 2001
Project Status Completed (Fiscal Year 2001)
Budget Amount *help
¥1,200,000 (Direct Cost: ¥1,200,000)
Fiscal Year 2001: ¥500,000 (Direct Cost: ¥500,000)
Fiscal Year 2000: ¥700,000 (Direct Cost: ¥700,000)
KeywordsTopological group / Free topological group / k-space / Quotient mapping / Metric space / 距離化可能空間 / フレッシェ空間 / ススリンプロパティー
Research Abstract

We could make progress in studying of topological structures of the free topological group F(X) on a metrizable space X. For each natural number n, let F_n(X) be a subspace of F_(X) formed by all words whose length is less than or equal to n. Then F_n(X) is a continuous image by the natural mapping i_n from the space (X 【symmetry】 X^<-1>【symmetry】{e})^n, where e is the unit element of F(X). It is well-known that the topology of F(X) has a complicated structure. In fact, the free topological group on a compact metric space, for example a convergent sequence with its limit, is not first countable, and hence not metrizable. On the other hand, it is known that if a space X is a compact metrizable space, then each F_n(X) is metrizable. In 2000, we obtained a necessary and sufficient condition of a metrizable space X such that each F_n(X) is metrizable. In 2001, we got an answer to an old problem "Find a necessary and sufficient condition of a space X such that each natural mapping i_n is a quotient mapping", when X is metrizable. Furthermore, as an application of the result, we could characterize a metrizable space X such that the free group topology of F(X) has a simple description such as a subset U of F(X) is open if and only if i_n^<-1>(U ∩ F_n(X)) is open in (X 【symmetry】 X^<-1> 【symmetry】 {e})^n for each natural number n.

Report

(3 results)
  • 2001 Annual Research Report   Final Research Report Summary
  • 2000 Annual Research Report
  • Research Products

    (20 results)

All Other

All Publications (20 results)

  • [Publications] Kohzo Yamada: "Frechet-Urysohn spaces in free topological groups"Proceedings of the American Mathematical Society.

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Kohzo Yamada: "Metrizable subspaces of free topological groups on metrizable spaces"Topology Proceedings. 23(1998). 379-409 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Haruto Ohta: "Extension properties and the Niemytzki plane"Applied General Topology. 1. 45-60 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Jin Ono: "The unique midpoint property of a subspace of the real line"Topology and its Applications. 104. 215-226 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Dmitri Shakhmatov: "Baire isomorphisms at the first level and dimension"Topology and its Applications. 107. 153-159 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Hiroshi Fujita: "A characterization of compactly generated metric groups"Proceedings of the American Mathematical Society.

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Kohzo Yamada: "Frechet-Urysohn spaces in free topological groups"Proceedings of the American Methematical Society. (in print).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Kohzo Yamada: "Metrizable subspaces of free topological groups on metrizable spaces"Topology Proceedings. 23(1998). 379-409 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Haruto Ohta: "Extension properties and the Niemytzki plane"Applied General Topology. 1. 45-60 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Jin Ono: "The unique midpoint property of a subspace of the real line"Topology and its Applications. 104. 215-226 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Dmitri Shakhmatov: "Baire isomorphisms at the first level and dimension"Topology and its Applications. 107. 153-159 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Hiroshi Fujita: "A characterization of compactly generated metric groups"Proceedings of the American Mathematical Society. (in print).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Kohzo Yamada: "Frechet-Urysohn spaces in free topological groups"Proceedings of the Americal Mathematical Society.

    • Related Report
      2001 Annual Research Report
  • [Publications] Kohzo Yamada: "Metrizable subspaces of free topological groups on Metrizable spaces"Topology Proceedings. 23(1998). 379-409 (2000)

    • Related Report
      2001 Annual Research Report
  • [Publications] Haruto Ohta: "Extension properties and the Niemytzki plane"Applied General Topology. 1. 45-60 (2000)

    • Related Report
      2001 Annual Research Report
  • [Publications] Dmitri Shakhmatov: "Baire isomorphisms at the first level and dimension"Topology and its Applications. 107. 153-159 (2000)

    • Related Report
      2001 Annual Research Report
  • [Publications] Hiroshi Fujita: "A characterization of compactly generated metric groups"Proceedings of the American Mathematical Society.

    • Related Report
      2001 Annual Research Report
  • [Publications] Kohzo Yamada: "Metrizable subspaces of free topological groups on metrizable spaces"Topology Proceedings. 23. 379-409 (1998)

    • Related Report
      2000 Annual Research Report
  • [Publications] Vladimir Pestov: "Free topologizal groups on metrizable spaces and inductive limits"Topology and its Applications. 98. 291-301 (1999)

    • Related Report
      2000 Annual Research Report
  • [Publications] Kohzo Yamada: "Frechet-Urysohn spaces in free topological groups"Proceedings of the American Mathematical Society.

    • Related Report
      2000 Annual Research Report

URL: 

Published: 2000-04-01   Modified: 2021-04-07  

Information User Guide FAQ News Terms of Use Attribution of KAKENHI

Powered by NII kakenhi