Descriptive Set Theoretical Studies of the Function Space of Irrationals
Project/Area Number 
16540098

Research Category 
GrantinAid for Scientific Research (C)

Allocation Type  Singleyear Grants 
Section  一般 
Research Field 
General mathematics (including Probability theory/Statistical mathematics)

Research Institution  Yokohama National University 
Principal Investigator 
TAMANO Kenichi Yokohama National University, Graduate School of Engineering, Professor (90171892)

CoInvestigator(Kenkyūbuntansha) 
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)

Project Period (FY) 
2004 – 2007

Project Status 
Completed (Fiscal Year 2007)

Budget Amount *help 
¥2,810,000 (Direct Cost: ¥2,600,000、Indirect Cost: ¥210,000)
Fiscal Year 2007: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2006: ¥600,000 (Direct Cost: ¥600,000)
Fiscal Year 2005: ¥600,000 (Direct Cost: ¥600,000)
Fiscal Year 2004: ¥700,000 (Direct Cost: ¥700,000)

Keywords  function space / topology / descriptive set theory / irrationals / topological space / 国際研究者交流 / フランス:米国 
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 M3space. 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 Mlspace or not, i.e., whether it has a sigmaclosurepreserving base or not. First, we tried to determine which kinds of properties does a sigmaclosurepreserving base have if it exists. Finally, with the aid of discussion with Gruenhage (a cooperative researcher), we proved that Ck(P) is an Mlspace, 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 Mlspace or not. As byproducts 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.

Report
(5 results)
Research Products
(10 results)