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2016 年度 実施状況報告書

コンパクト型可換群の構造及びMarkov稠密性を実現する群位相の導入の研究

研究課題

研究課題/領域番号 26400091
研究機関愛媛大学

研究代表者

D・B Shakhmatov  愛媛大学, 理工学研究科(理学系), 教授 (90253294)

研究期間 (年度) 2014-04-01 – 2018-03-31
キーワードselective property / sequentially compact / omega-bounded group / weakly pseudocompact / pseudocompact / completeness properties / direct products / direct sums
研究実績の概要

We introduce the new class of selectively sequentially pseudocompact spaces. We say that a topological space X is selectively sequentially pseudocompact (SSP for short) if for every sequence (U_n) of non-empty open subsets of X, one can choose a point x_n in U_n for every n in such a way that the sequence (x_n) has a convergent subsequence. The class of SSP spaces is closed under taking arbitrary products and continuous images, contains the class of all dyadic spaces and forms a proper subclass of the class of strongly pseudocompact spaces. We prove that every omega-bounded (=the closure of which countable set is compact) group is SSP, while compact spaces need not be SSP.
(2)We prove that many completeness properties coincide in metric spaces, precompact groups and dense subgroups of products of separable metric groups. We apply these results to function spaces C_p(X,G) of G-valued continuous functions on a space X with the topology of pointwise convergence, for a separable metric group G. A space X is weakly pseudocompact if it is G_delta-dense in at least one of its compactifications. A topological group G is precompact if it is topologically isomorphic to a subgroup of a compact group. We prove that every weakly pseudocompact precompact topological group is pseudocompact, thereby answering positively a question of Tkachenko.
(3) We study subgroups of direct products closely approximated by direct sums.
In paticular, we study uniformly controllable, controllable and weakly controllable subgroups of direct products and their connections with coding theory.

現在までの達成度 (区分)
現在までの達成度 (区分)

3: やや遅れている

理由

We discovered a new proper subclass of the class of pseudocompact spaces containing the class of sequentially compact spaces which is closed under arbitrary products and continuous images. In order to keep the priority of our discovery, we thouroughly investigated the properties of this new class, especially in case of topological groups. As a side effect, we have also solved a problem if Garcia-Ferreira and Tomita. We believe that the new class of spaces will prove to be important in our understanding of the structure of pseudocompact spaces and groups.

今後の研究の推進方策

We shall continue our investigation of selective sequential pseudocompactness, as well as related selective properties introduced by means of topological games. We shall investigate the realization of the Zariski closure of an abelian group G by the closure in some pseudocompact group topology on G.

  • 研究成果

    (10件)

すべて 2017 2016

すべて 雑誌論文 (3件) (うち国際共著 3件、 査読あり 3件、 謝辞記載あり 2件、 オープンアクセス 1件) 学会発表 (7件) (うち国際学会 4件、 招待講演 4件)

  • [雑誌論文] Subgroups of direct products closely approximated by direct sums2017

    • 著者名/発表者名
      M.V. Ferrer, S. Hernandez, D. Shakhmatov
    • 雑誌名

      Forum Mathematicum

      巻: 29 ページ: 1125-1144

    • DOI

      10.1515/forum-2016-0047

    • 査読あり / 国際共著
  • [雑誌論文] Selective sequential pseudocompactness2017

    • 著者名/発表者名
      Alejandro Dorantes-Aldama and Dmitri Shakhmatov
    • 雑誌名

      Topology and its Applications

      巻: 222 ページ: 53-69

    • DOI

      10.1016/j.topol.2017.02.016

    • 査読あり / 国際共著 / 謝辞記載あり
  • [雑誌論文] Completeness and compactness properties in metric spaces, topological groups and function spaces2017

    • 著者名/発表者名
      Alejandro Dorantes-Aldama and Dmitri Shakhmatov
    • 雑誌名

      Topology and its Applications

      巻: - ページ: -

    • DOI

      10.1016/j.topol.2017.04.012

    • 査読あり / オープンアクセス / 国際共著 / 謝辞記載あり
  • [学会発表] Completeness properties in topological groups2016

    • 著者名/発表者名
      A. Dorantes-Aldama
    • 学会等名
      2016年度ジェネラルトポロジーシンポジウム
    • 発表場所
      筑波大学(つくば市)
    • 年月日
      2016-12-07 – 2016-12-09
  • [学会発表] Selective sequential pseudocompactness in topological groups2016

    • 著者名/発表者名
      A. Dorantes-Aldama
    • 学会等名
      RIMS研究集会「集合論的・幾何学的トポロジーとその応用」
    • 発表場所
      京都大学数理解析研究所(京都府京都市)
    • 年月日
      2016-10-17 – 2016-10-19
  • [学会発表] New class of pseudocompact spaces containing all sequentially compact spaces2016

    • 著者名/発表者名
      D. Shakhmatov
    • 学会等名
      Alexandroff Topology Seminar at Moscow State University
    • 発表場所
      Moscow (Russia)
    • 年月日
      2016-09-22 – 2016-09-22
    • 招待講演
  • [学会発表] Completeness properties in metric spaces and topological groups, with applications to function spaces2016

    • 著者名/発表者名
      D. Shakhmatov
    • 学会等名
      IMAC Topology Workshop
    • 発表場所
      Universitat Jaume I, Castellon (Spain)
    • 年月日
      2016-09-16 – 2016-09-19
    • 国際学会 / 招待講演
  • [学会発表] Selective sequential pseudocompactness in topological spaces and groups2016

    • 著者名/発表者名
      D. Shakhmatov
    • 学会等名
      Interdisciplinary Colloquium in Topology
    • 発表場所
      University of Navarra, Pamplona (Spain)
    • 年月日
      2016-09-01 – 2016-09-02
    • 国際学会 / 招待講演
  • [学会発表] Selectively sequentially pseudocompact group topologies on abelian groups2016

    • 著者名/発表者名
      D. Shakhmatov
    • 学会等名
      Twelfth Symposium on General Topology and its Relations to Modern Analysis and Algebra (TOPOSYM 2016)
    • 発表場所
      Prague (Czech Republic)
    • 年月日
      2016-07-25 – 2016-07-29
    • 国際学会
  • [学会発表] Productively Baire completeness and compactness properties2016

    • 著者名/発表者名
      D. Shakhmatov
    • 学会等名
      2016 International Conference of the Honam Mathematical Society
    • 発表場所
      Chonbuk National University (Deparment of Mathematics Education), Jeonju, South Korea
    • 年月日
      2016-06-16 – 2016-06-18
    • 国際学会 / 招待講演

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公開日: 2018-01-16  

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