2001 Fiscal Year Final Research Report Summary
A study of continuous selections for filter spaces
| Project/Area Number |
12640129
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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 |
General mathematics (including Probability theory/Statistical mathematics)
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| Research Institution | Ehime University |
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Principal Investigator |
NOGURA Tsugunori Ehime University, Faculty of Sciences, Professor, 理学部, 教授 (00036419)
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| Co-Investigator(Kenkyū-buntansha) |
HATTORI Yasunao Shimane University, Faculty of Sciences and Engineerings, Professor, 理工学部, 教授 (20144553)
FUJITA Hiroshi Ehime University, Faculty of Sciences, Instructor, 理学部, 助手 (60238582)
SHAKHMATOV Dmitri Ehime University, Faculty of Sciences, Professor, 理学部, 教授 (90253294)
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| Project Period (FY) |
2000 – 2001
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| Keywords | Hyperspace / Selection / Vietoris topology / Fell Topology |
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| Research Abstract |
Let X be a topological space. We denote by 2^X the collection of non-empty closed subsets. The set 2^X with the Vietoris topologyis called hyperspace. A map σ : 2^X → X is called a selection if σ(F) ∈ F for every F ∈ 2^X. We characterize various topological properties which admit continuous selections. It is known that if 2^X admits a continuous selection, then X is hereditarily Baire. Using this fact we have shown:
(1) A countable regular space admits a continuous selection if and only if it is scattered.
Also we have shown:
(2) A Hausdorff space admits a Fell continuous selection if and only if it is topological well-orderable.
Let κ be cardinal and let p be a filter on κ. By κ(p) we denote the space which is discrete at points of κ and a neighborhood base of p is given by the fomular {F ∪ {p} : F ∈ p}. For these type of spaces we have the following results:
(3) If p has a nested base, then κ(p) admits a continuous selection.
(4) A co-countable filter on ω_1 admits a continuous selection but not on ω_2.
(5) Let p_1 be a filter on κ_1 with a nested base and p_2 be a filter on ω. If the sum of κ(p_1) 【symmetry】 ω(p_2) admits a continuous selection, then p_2 is the Frechet filter.
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