2003 Fiscal Year Final Research Report Summary
Research of dimension and topological structures of topological groups
| Project/Area Number |
14540071
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Geometry
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| Research Institution | Shizuoka University |
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Principal Investigator |
YAMADA Kohzo Shizuoka University, Faculty of Education, Department of Mathematics, Professor, 教育学部, 教授 (00200717)
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| 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)
SAKAI Masami Kanagawa University, Faculty of Engineering, Professor, 工学部, 教授 (60215598)
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| Project Period (FY) |
2002 – 2003
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| Keywords | Topological group / Free topological group / k-space / Quotient mapping / Metric space / Tightness |
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| Research Abstract |
Let F(X) and A(X) be the free topological group and the free abelian topological group on a topological space X, respectively. For each natural number n, F_n(X) denotes a subspace of F(X) consisting of all words whose length are less than or equal to n. In the abelian case, we. use A_n(X) to denote the same subspace of A(X). Then, each 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).
In 1989, Arhangel'skii, Okunev and Pestov asked whether for a metric space X, the tightness of A(X) is equal to the weight of the space of all non-isolated points of X. In this research, we proved that the answer is affirmative under the set-theoretic axiom V=L. Furthermore, in the non-abelian case, we proved that for a metric space X, the tightness of F(X) is equal to the weight of X under the set-theoretic axiom V=L.
For last 3 years, we got an answer to an old problem "Find a necessary and sufficient condition of a space X such that each mapping i_n is a quotient mapping", when X is metrizable. That is, we characterized a metrizable space X such that the mapping i_n is a quotient mapping for each natural number n for both F(X) and A(X), respectively.
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