Descriptive Set Theoretical Studies of the Function Space of Irrationals

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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.