2016 Fiscal Year Research-status Report
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
| Project/Area Number |
26400091
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| Research Institution | Ehime University |
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Principal Investigator |
D・B Shakhmatov 愛媛大学, 理工学研究科(理学系), 教授 (90253294)
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| Project Period (FY) |
2014-04-01 – 2018-03-31
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| Keywords | selective property / sequentially compact / omega-bounded group / weakly pseudocompact / pseudocompact / completeness properties / direct products / direct sums |
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| Outline of Annual Research Achievements |
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.
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| Current Status of Research Progress |
Current Status of Research Progress
3: Progress in research has been slightly delayed.
Reason
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.
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| Strategy for Future Research Activity |
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.
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Research Products
(10 results)
