| General Topology has gone through over one hundred years' development. Now its theories,spirits and methods have been seeped into almost every important field of mathematics even into physics,chemistry,biology and engineering. So far,mathematicians who study Topology have obtained substantial achievements in important topological directions,such as generalized metric space,cardinal function,compactness,dimension theory,etc. But what is worth to paying attention to is that we often draw on one special type of topological space to think and solve problems,for example,the classical structures,Sorgenfrey line K,Michael line RQ,Niemytzki planeTV,KxK,#Q xP,etc.;subtle and profound topological properties aredetailedly characterized by them. This type of Topological space is GO-space or their products. The special ordinal structure of GO-space provide people extensive space to imagine and plot. Not only does GO-space provide rich examples,but also GO-space buildes a bridge between general topology and related mathematics branches,such as Lattics theory,Domain theory,Graph theory,Real number theory,etc. Thus it is very important in theory and reality to study GO-space. Reseachers who study Topology have preliminarily explored GO-spaces in early twentieth century. Not only have the reseach results about GO-space enriched the contents of general topology,but also have increased ingenious creativity and breakthrough and made general topology more wonderful and lively because of making use of the theory about ordinal and cardinal in set theory,axiomitic system and combinatorial theory.In this paper,the best contribution is to discover that the cofinalities of elements of minimal linearly compactification of GO-space exactly reflect the properties of GO-space. Thus we can set up the relation between GO-space and ordinals and resolve some problems which are not resolved by predecessors.In 1971,Lutzer proved that every GO-space is countably paracompact. But we know it is generally not paracompact. In the first section of this paper,we will prove a sufficient and necessary condition about GO-space being K-paracompact (where K is a regular cardinal) and therefore we can deduce a sufficient and necessary condition about GO-space being paracompact and the well knwon result that every GO-space is countably paracompact.The following result was obtained in 1983:K is < K-paracompact,where K is a regular uncountable cardinal and 77 a cadinal satisfing 77 < 玔2]. In the second section,we will generalize this to the product of GO-spaces. Our result is as follows:Let {Xa}an and a w,then IIaK(i -cfx>K).In the other part of this section,we will define point-shrinking property and prove that the product of two ordinals p. v has < min{cfu,cfv} point-shrinking property.In 2000,N.Kemoto,K.Tamano and Y.Yajima proved sufficient and necessary conditions about subspace in product of two ordinals being paracompact,metacompact,screenable,etc.[14]. In 1992,N.Kemoto,H.Ohta and K.Tamano proved a sufficient and necessary condition about the normality of subspaces in product of two ordinals[3];in 1996,N.Kemoto,T.Nogura,K.D.Smith,and Y.Yajima proved that for every subspace of product of two ordinals,the normality,OWN,shrinking property are equivalent [11];but all of these don't break the peculiarity of ordinals. In the third,fourth and fifth section of this paper,we will genenralize these to the subspaces of the product of two GO-spaces.Two main results of the third section are as follows:1. Let X,Y be two GO-spaces,and every subspace of Xw is not dense in some closed subspace of LX and every subspace of Yw is not dense in some closed subspace of Then Z is paracompact if and only if for every (x,y),the following holds:(1). If for some i . 2,i - cfx = u,> LJ holds,then there exists a closed unbounded subse...
|
PDF Full Text Request