A study of continuous selections for filter spaces

过滤空间连续选择的研究

• 批准号: 12640129
• 负责人: NOGURA Tsugunori
• 金额: $ 1.6万
• 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: 2000
• 资助国家: 日本
• 起止时间: 2000 至 2001
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-12640129/
• 关键词: Hyperspace; Selection; Vietoris topology; Fell Topology; Vietoris空間

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.

令X为拓扑空间。我们用2^x表示非空封闭子集的集合。带有越野拓扑的2^x的集合称为Hyperspace。如果每个f∈2^x x（f）∈F，则映射σ：2^x→x被称为选择。我们表征了接受连续选择的各种拓扑特性。众所周知，如果2^x接受连续的选择，则x是遗传性的。使用这个事实，我们已经显示了：（1）可计数的常规空间在且仅当散布时接受连续的选择。​​此外，我们已经表明：（2）Hausdorff空间在且仅当它是拓扑良好的情况下，就可以接下来是连续的选择。​​LETκ是红衣主教，并在κ上进行p p filter pheκ。通过κ（P），我们表示在κ点离散的空间，P的邻域基础由Fomular {f∪{p}给出：f∈P}。对于这些类型的空间，我们有以下结果：（3）如果P具有嵌套基碱基，则κ（P）承认连续选择。（4）ω_1上的可共计过滤器承认连续选择，但在ω_2上不接受。（5）让P_1在κ_1上使用κ_1，用嵌套的基础和P_2是一个过滤器。如果κ（p_1）[对称性]ω（P_2）的总和接收连续选择，则P_2是Frechet滤波器。

项目成果

S.Fujii, K.Miyazaki, T.Nogura: "Vietoris continuous selections on scattered spaces"Journal of the Mathematical Society of Japan. (印刷中). (2002)

S.Fujii、K.Miyazaki、T.Nogura：“离散空间上的 Vietoris 连续选择”日本数学会杂志（2002 年出版）。

V.Gutev, T.Nogura: "Selections for Vietoris-like hyperspace topologies"Proceedings of London Mathematical Society. 80. 235-256 (2000)

V.Gutev、T.Nogura：“类 Vietoris 超空间拓扑的选择”伦敦数学会论文集。

V. Gutev, S. Garcia-Ferreira, T. Nogura and others: "Extreme selections for hyperspaces of topological spaces"Topology Appl.. (accepted).

V. Gutev、S. Garcia-Ferreira、T. Nogura 等人：“拓扑空间超空间的极端选择”拓扑应用程序..（已接受）。

V.Gutev, T.Nogura: "Selections and order-like relations"Applied general topology. 2. 205-218 (2001)

V.Gutev、T.Nogura：“选择和类序关系”应用一般拓扑。

V.Gutev, S.Garcia-Ferreira, T.Nogura, others: "Extreme selections for hyperspaces of topological spaces"Topology Appl. (印刷中).

V. Gutev、S. Garcia-Ferreira、T. Nogura 等：“拓扑空间超空间的极端选择”拓扑应用（正在出版）。