2007 Fiscal Year Final Research Report Summary
Descriptive Set Theoretical Studies of the Function Space of Irrationals
| Project/Area Number |
16540098
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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 | Yokohama National University |
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Principal Investigator |
TAMANO Kenichi Yokohama National University, Graduate School of Engineering, Professor (90171892)
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| Co-Investigator(Kenkyū-buntansha) |
TERADA Toshiji Yokohama National University, Graduate School of Environment and Information Sciences, Professor (80126383)
SHIOJI Naoki Yokohama National University, Graduate School of Environment and Infnrmation Srianens, Associate Professor (50215943)
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| Project Period (FY) |
2004 – 2007
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| Keywords | function space / topology / descriptive set theory / irrationals / topological space |
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| Research Abstract |
Let P be the space of irrationals with the usual topology, and Ck(P) be the space of all real valued functions on P with the compact open topology. In 1961, Ceder raised the question whether every M3 space is an M1 or not., which is called the M3=> M 1 question. In 2000, Gartside and Reznichenko showed that Ck(P) is an M3-space. After then, it had been conjectured that Ck(P) can be a candidate of a counterexample for the M3 => M 1 question. Gartside, Gruenhage, Nyikos and Tamano had studied that. The purpose of this research was to determine whether Ck(P) is an Ml-space or not, i.e., whether it has a sigma-closure-preserving base or not.
First, we tried to determine which kinds of properties does a sigma-closure-preserving base have if it exists. Finally, with the aid of discussion with Gruenhage (a cooperative researcher), we proved that Ck(P) is an Ml-space, by using a method by Mizokami and Shimane, and by using the fact that Ck(P) is of the first category, which completes the main purpose of our research.
But still the M3=> M 1 question. is open. For example it is unknown whether every subspace of Ck(P) is an Ml-space or not. As by-products of our research, we obtained several new constructions of bases and a monotone normality operator of Ck(P), which might be helpful for further research.
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