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Algebraic structure of compact-like topological groups and convergence properties

紧类拓扑群的代数结构及收敛性

基本信息

批准号：
15540082
负责人：
SHAKHMATOV D.B
金额：
$ 2.3万
依托单位：
Ehime University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2003
资助国家：
日本
起止时间：
2003 至 2005
项目状态：
已结题

项目摘要

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.

设X是拓扑群G的子空间。我们说X拓扑生成G，只要由X代数生成的G的最小子群在G中稠密。在G的所有拓扑生成G的闭子集X中，存在一个具有最小可能权w（X）的子集，我们称这个权为拓扑生成权。本文研究了紧群G的拓扑生成权，得到了如下结果：定理1.零维紧Abel群G的拓扑生成权与G的权重合.定理2.连通紧Abel群G的拓扑生成权与G的权的ω-根重合. (Here基数k的Ω根是使得s的Ω幂超过k的最小可能基数s。定理3.紧Abel群G的拓扑生成权等于G的连通分支c（G）的拓扑生成权与G/c（G）的权的乘积，并研究了可数紧Abel群的代数结构。特别地，我们研究了一个大小至多为2^c的阿贝尔群G是否允许一个可数紧群拓扑。(Here c表示连续统的基数。）定理4.对于阿贝尔群G，下列条件是等价的：（i）G容许可分可数紧群拓扑，（ii）G容许遗传可分可数紧群拓扑，（iii）G容许遗传可分可数紧群拓扑，（iv）G的大小至多为2^c，满足条件Ps和CC.定理5.对于无穷阿贝尔群，下列条件是等价的：（i）G有可分伪紧群拓扑，（ii）G的基数在c和2^c之间，且满足条件Ps.

项目成果

期刊论文数量（48）

专著数量（0）

科研奖励数量（0）

会议论文数量（0）

专利数量（0）

Selection pointwise maximal spaces

选择逐点最大空间

DOI：
发表时间：
2005
期刊：
Topology and its Applications 146
影响因子：
0
作者：
V.Gutev;T.Nogura
通讯作者：
T.Nogura

Homogeneous systems of parameters in cohomology algebras of finite groups.

有限群上同调代数中的齐次参数系统。

DOI：
发表时间：
2004
期刊：
Archiv der Mathematik (Basel) 82
影响因子：
0
作者：
SASAKI;Hiroki
通讯作者：
Hiroki

Extensions of 2-point selections

2点选择的扩展

DOI：
发表时间：
2008
期刊：
New Zealand Journal of Mathematics 38
影响因子：
0
作者：
S. Dolecki;H. Kunzi;T. Nogura;Toru Ohmoto;Jun O' Hara (今井の旧姓);D. Dikranjan; D. Shakhmatov;Jun O' Hara (今井の旧姓);K. Miyajima;Remi Langevin and Jun O'Hara;D. Dikranjan; D. Shakhmatov;H. Sainohira and Shoji Tsuboi;今井淳;S. Garcia-Ferreira; V. Gutev; T. Nogura
通讯作者：
S. Garcia-Ferreira; V. Gutev; T. Nogura

Around monothetic groups

围绕单一群体