Descriptive Set Theoretical Studies of the Function Space of Irrationals

设P是具有通常拓扑的无理数空间，Ck(P)是P上具有紧开拓扑的所有实值函数的空间。1961年，Ceder提出了是否每个M3空间都是M1的问题，称为M3=&gt；M1问题。2000年，Gartside和RezNichenko证明了Ck(P)是一个M3空间。此后，有人猜测Ck(P)可能是M3=&gt；M1问题的反例的候选者。加特赛德、格伦海奇、尼科斯和塔马诺对此进行了研究。本研究的目的是判断Ck(P)是否为一个ML-空间，即它是否有一个sigma-闭包保基。首先，我们试图确定一个sigma-闭包保基如果存在，它具有哪些性质。最后，利用Mizokami和Shimane的方法，并利用Ck(P)是第一类的事实，借助与Gruenage(合作研究者)的讨论，我们证明了Ck(P)是一个ML-空间，从而完成了我们的主要研究目的。是开放的。例如，Ck(P)的每个子空间是否都是ML-空间是未知的。作为我们研究的副产品，我们得到了几个新的基的构造和Ck(P)的一个单调正规性算子，这可能对进一步的研究有所帮助。

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.