The considerations of M_3 problems from a new point of view

M_3-空间类似乎是包含度量空间和CW-复形的最重要的类之一。与我们的结果密切相关的M_3-空间问题是具有一个M_3因子的乘积的正规性与可数仿紧性的等价性问题。这个问题于1992年提交，至今仍未解决。然而，我们已经认识到它有必要研究具有M3因子的乘积的覆盖性质。2003年，我们开始研究这类乘积的基仿紧性这一新概念。这一概念被认为有些特殊。2004年，这项研究突然发展成为积的正规覆盖刻画的一般论点。其中的一个基本思想是对产品使用矩形细化的概念。2005年，我们证明了无限乘积的正规覆盖的相同刻画。一般来说，有限乘积和无限乘积是如此不同，他们在那里使用的技术也有很大的不同。因此，这两种产品都获得了相同的结果，这应该是令人惊讶的。此外，利用我们所使用的技巧，我们可以肯定地回答1989年提出的关于Σ乘积的正规性的问题之一。Nobuyuki Kemoto主要研究了序数的两个子空间的乘积的正规性和一些覆盖性质。它们是如此的特殊，以至于有可能在这个世界上获得许多令人惊讶的结果。酒井雅美也得到了许多关于函数空间和自由拓扑群的结果，这些结果将在不久的将来与上述结果相关。

The class of M_3-spaces seems to be one of the most important classes which contain metric spaces and CW-complexes. The problem of M_3-spaces which is deeply related to our results is that of the equivalence of normality and countable paracompactness of products with one M_3 factor.This problem was submitted in 1992 and it is still open. However, we have recognized that it is necessary for it to study the covering properties of products with an M3 factor. In 2003, we began to study the new concept called base-paracompactness of such products. This concept was considered rather somewhat special. In 2004, this study was suddenly, developed as the general arguments of characterizations for normal covers of products. An essential idea in there is to use the concept of rectangular refinements for products. In 2005, we proved the same characterizations for normal covers of infinite products. Generally, finite products and infinite products are so different that their technique used there are also quite different. So it should be surprising to obtain the same results for both products. Moreover, using our techniques used here, we could give an affirmative answer to one of the problems concerning the normality of Σ-products, which was raised in 1989.Nobuyuki Kemoto has been mainly studied the normality and some covering properties of the products of two subspaces of ordinals. They axe so special that it is possible to obtain many surprising results in this world. Masami Sakai has also obtained many results for function spaces and free topological groups, which will be related to the above results in the near future.