1999 Fiscal Year Final Research Report Summary
Set Theoretic Approaches for Normality of Product Spaces
| Project/Area Number |
09640122
|
|---|
| Research Category |
Grant-in-Aid for Scientific Research (C)
|
| Allocation Type | Single-year Grants |
|---|
| Section | 一般 |
|---|
| Research Field |
Geometry
|
|---|
| Research Institution | Oita University |
|---|
Principal Investigator |
KEMOTO Nobuyuki Oita Univ. Education and Welfare Science, Professor, 教育福祉科学部, 教授 (70161825)
|
|---|
| Co-Investigator(Kenkyū-buntansha) |
MORI Naganori Oita Univ. Education and Welfare Science, Professor, 教育福祉科学部, 教授 (40040737)
BABA Kiyoshi Oita Univ. Education and Welfare Science, Professor, 教育福祉科学部, 教授 (80136770)
KITA Hiroo Oita Univ. Education and Welfare Science, Professor, 教育福祉科学部, 教授 (20224941)
OGATA Takehide Oita Univ. Education and Welfare Science, Associate Professor, 教育福祉科学部, 助教授 (90037268)
|
|---|
| Project Period (FY) |
1997 – 1999
|
| Keywords | Normal / Product / Countably paracompact / Orthocompact / Paracompact / V=L / PMEA |
|---|
| Research Abstract |
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
|
