单调空间与度量化定理

On Monotone Spaces and Metrization Theorems

本文回答了关于MCM空间遗传性的一个问题,讨论了k-MCM空间是k半层空间的条件,得到了一些用g函数刻划的度量化定理.主要结论有:MCM空间是关于Fσ子空间遗传的;在正规空间类中,q空间(ωN空间,k-MCM空间)是关于开Fσ子空间遗传的;如果X是具有Gδ对角线的正则次中紧 k-MCM空间,则X是k半层空间;X是可度量化空间的充要条件是存在X上的g函数满足对X中任意不相交的闭集F与紧集C,都有某个n∈ω,使得(∪x∈F g(n,x))∩(∪y∈C g(n,y))=(?).

In this paper an open problem about the hereditary of MCM-spaces is an-swered affirmatively, the condition which k-MCM-spaces are k-semistratifiable spaces is discussed, and some metrizable theorems are obtained by g-functions. The main results are that MCM-spaces are hereditary with respect to Fσ-subspaces; q-spaces(ωN-spaces, k-MCM-spaces) are hereditary with respect to open and Fσ-subspaces in normal spaces; a regular submesocompact k-MCM-space with a Gδ-diagonal is k-semistratifiable; and a space X is metrizable if and only if there is a g-function on X such that for any closed set F and compact set C in X, if