Set Theoretic Approaches for Normality of Product Spaces

产品空间正态性的集合理论方法

基本信息

批准号：
09640122
负责人：
KEMOTO Nobuyuki
金额：
$ 1.09万
依托单位：
Oita University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1997
资助国家：
日本
起止时间：
1997 至 1999
项目状态：
已结题

项目摘要

In this research project, we proved on countable metacompactness:(1) An subspaces of αィイD12ィエD1 are countably metacompact for suitably large α.(2) An subspaces of ωィイD3η(/)1ィエD3 are countably metacompact for every ηεω.(3) There is a subspace of ωィイD3ω(/)1ィエD3 which is not countably metacompact.After then we obtained an affirmative answer of a problem on normality raised by Kemoto, Ohta and Tamono, that is, normality and collectionwise normality an equivalent for all subspaces ofαィイD12ィエD1. We proved: For every subspace X of ωィイD32(/)1ィエD3,(1) X is normal iff X is apandable iff X is coutably paracompact and strongly collectionwise Hausdorff.(2) If V=L or the Product Measure Extension Axiom are assumed, then X is normal iff X is countably paracompact.(3) X is collectionwise Hausdorff.We have also investigated on orthocompactness: In the realm of subspaces of products of two ordinals:(1) Orthocompactness and weak suborthocompactness are equivalent.(2) There is an orthocompact subspace of … More ωィイD32(/)1ィエD3 which is not normal.(3) Normal subspaces of ωィイD32(/)1ィエD3 are orthocompact.(4) There is a normal subspace of (ωィイD21ィエD2+1)ィイD12ィエD1 which is not orthocmpact.(5) If X is a subspace of ωィイD21ィエD2×ωィイD22ィエD2 such that X∩(α+1)×ωィイD22ィエD2 and X∩ωィイD21ィエD2×(β+1) are orthocompact for each α<ωィイD21ィエD2 and β<ωィイD22ィエD2, then X is orthocompact.Around paracompactness, we proved the following results: Let consider subspaces of products of two ordinal:(1) For such subspaces, weak submetaLindelofness and metacompactness are equivalent.(2) For such subspaces, subparacompactness implies metacompactness.(3) Metacompact subspaces ofωィイD32(/)1ィエD3 are paracompact.(4) Metacompact subspaces of ωィイD32(/)2ィエD3 are subparacompact.(5) There is a metacompact subspace of (ωィイD21ィエD2+1)ィイD12ィエD1 which is not paracompact.(6) There is a metacompact subspace of (ωィイD22ィエD2+1)ィイD12ィエD1 which is not subparacompact.On sequential completeness, we proved: let κ be a cardinal number with the usual order topology:(1) All subspaces of κィイD12ィエD1 are weakly sequentially compete.(2) All subspaces of ωィイD32(/)1ィエD3 are sequentially complete.(3) There is a subspace of (ωィイD21ィエD2+2)ィイD12ィエD1 which is not sequentailly complete.(4) A×B is sequentially complete whenever A and B are subspaces of κ. Less

在该研究项目中，我们提供了可计数的元素化度：（1）α-D12 D1的子空间对于适当的大α的子空间是可计算的。（2）ω-D3η（/）1 D3的子空间在每个ηεΩ中都可以计数为元素。 Kemoto，Ohta和Tamono提出的正态性问题的答案，即正态性和收集正态性对于α-D12-E D1的所有子空间都相等。我们还研究了矫正性：在两个普通产品的子空间的领域中：（1）正交性和弱的亚螺旋体紧缩性等效。（2）有一个矫正的子空间……多ωiid32（/）d32（/）1ie d3的正式分子空间不是正常的。（3）正常的子空间。（ωid21ie d2+1）的正常子空间IE D12IE D1不是正交的。（5）如果x是ωd21ed2×ωd22ed2的子空间，则每个X+x∩（α+1）+1）×d2 d2 d2 d2 d2 d2 d2 d2 d21e d221e d221e d221e d221e+β+β+β+β（β） α<ωd21ed2和β<ωd22ed2，然后x是矫形的。对旁paracrocactnesmens，我们证明了以下结果：让考虑两个有序的产物的子空间：（1）对于此类子空间，弱的子类别性和元素性的较弱的亚物质性和元素性为subspact subspact。 ωd32（/）1ied3的子空间是副反应。（4）ωd32（/）2ied3的元合理子空间是subparaccompact。（5）有一个（ωd21ied2+1）IED12IED1的元元素子空间不是地下序列。 κD12E D1的所有子空间均具有弱依次的能力。（2）ωD32（/）1E D3的所有子空间均已顺序完成。（3）有一个（ωd21ed2+2）II D12E D1的子空间，均未顺序完成。较少的