2005 Fiscal Year Final Research Report Summary
Algebraic structure of compact-like topological groups and convergence properties
| Project/Area Number |
15540082
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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 | Ehime University |
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Principal Investigator |
SHAKHMATOV D.B Ehime University, Faculty of Science, Professor, 理学部, 教授 (90253294)
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| Co-Investigator(Kenkyū-buntansha) |
NOGURA Tsugunori Ehime University, Faculty of Science, Professor, 理学部, 教授 (00036419)
KISO Kazuhiro Ehime University, Faculty of Science, Professor, 理学部, 教授 (60116928)
SASAKI Hiroski Ehime University, Faculty of Science, Professor, 理学部, 教授 (60142684)
FUJITA Hiroshi Ehime University, Faculty of Science, Assistant, 理学部, 助手 (60238582)
YAMADA Kohzo Shizuoka University, Faculty of Education, Professor, 教育学部, 教授 (00200717)
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| Project Period (FY) |
2003 – 2005
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| Keywords | topological group / countably compact space / pseudocompact space / Abelian group / separable space / hereditarily separable space / weight / 位相群 |
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| Research Abstract |
Let X be a subspace of a topological group G. We say that X topologically generates G provided that the smallest subgroup of G algebraically generated by X is dense in G. Among all closed subsets X of G topologically generating G there exists one that has the smallest possible weight w(X), and we call this weight topologically generating weight. We investigated the topologically generating weight of a compact group G and obtained the following results :
Theorem 1.Topologically generating weight of a zero-dimensional compact Abelian group G coincides with the weight of G.
Theorem 2.Topologically generating weight of a connected compact Abelian group G coincides with the omega-root of weight of G. (Here the omega-root of a cardinal k is the smallest possible cardinal s such that the omega power of s exceeds k.)
Theorem 3.Topologically generating weight of a compact Abelian group G is equal to the product of the topologically generating weight of the connected component c(G) of G and the weight of G/c(G).
We also study algebraic structure of countably compact Abelian group. In particular, we investigate whether an Abelian group G of size at most 2^c admits a countably compact group topology. (Here c denotes the cardinality of the continuum.) Using forcing, we have constructed a model M of Zermelo-Fraenkel Axioms of Set Theory in which the following Theorm 4 holds.
Theorem 4.For an Abelian group G the following conditions are equivalent :
(i)G admits a separable countably compact group topology,
(ii)G admits a hereditarily separable countably compact group topology,
(iii)G admits a hereditarily separable countably compact group topology without infinite compact subsets,
(iv)G has size at most 2^c and satisfies conditions Ps and CC.
Theorem 5.For an infinite Abelian group the following conditions are equivalent :
(i)G has a separable pseudocompact group topology,
(ii)G has cardinality between c and 2^c and satisfies condition Ps.
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