【作者】 张丽；

【导师】 江守礼；

【作者基本信息】 山东大学 ， 基础数学， 2006， 硕士

【摘要】 Ceder于1961年发表了他的著名文章"Some generalizations of metric spaces"。在这篇文章中，Ceder引入了M_i^-空间（i=1，2，3），并试图给出M_i^-空间之间的关系，（i=1，2，3），其中M₁（?）M₂，M₂（?）M₃已经证明，但是在证明M₃（?）M₂，M₂（?）M₁时，遇到了困难，从而Ceder提出了M₃（?）M₂，M₂（?）M₁是否成立的经典性拓扑问题。从此这一问题吸引了很多拓扑学家的注意力。二十世纪七十年代，Gruenhage[1976]和Junnila[1978]分别采用不同的方法独立地证明了M_3^-空间是M_2^-空间。这一结果激励人们以更大的热情去研究：是否M_2^-空间也是M_1^-空间的问题。因为M_3-空间就是M_2-空间，我们通常把这一问题称为M₃（?）M₁问题。M.Ito[1983]在他的文章"Space whose closed images are M₁"中得到一个重要的结果：拓扑空间X的任一点x都有闭包保持局部基的层空间是M_1^-空间。Heath R.W.,Lutzer D.J.,Zenor P.L.[1973]在文章"Monotonically normal spaces"中引进一种很强的正规性-单调正规性，主要用以（连同半层空间）刻画层空间。Robert E.Buck[1996]在文章"Some weaker monotone separation and basis properties"中给出了M_i^-空间的局部化：m_i^-空间（i=1，2，3）与单调正规性的推广：单调T₂算子。在这篇文章中，Buck通过研究m_i^-空间（i=1，2，3）的某些性质，发现了m_i^-空间（i=1，2，3）与单调T₂空间的联系。除了这些，Buck也得到了单调正规的加强：ayclic单调正规和强单调正规。 我们知道，网是基的推广，由此可以得到很多重要的拓扑空间。因此我们从网的角度推广了m_1^-空间，定义得到了σ_1^-空间，并研究获得关于σ_1^-空更多还原

【Abstract】 In Ceder’s famous paper " some generations of metric spaces "which was published in 1961,he first studied the concepts of M_i-spaces,i=1,2,3 and tried to discover the relationships between M_i-spaces,i=1,2,3,which has been proved that M_i （?） M₂,M₂ （?） M₃ is right,but the opposite’s is still open.Since then many topologists have devoted their attention to the question on M₃（?） M₂, M₂（?） M₁.In 1970’,Gruenhage[1976]and Junnila[1978] used different methods to prove that M₃-spaces are M₂ -spaces independently. Their results inspirited the more enthusiasm of person,who want to solve the problem whether the M₂-spaces are Mi-spaces .M.Ito [1983]has proved that stratifiable spaces in which there is a closure preserving local base at each point are M₁ -spaces in his paper " Space whose closed images are ". Robert E.Buck[1996]studied m_i-spaces,locallized versions of Ceder’s Mi-spaces inspired by Ito’s theorem in his paper " Some weaker monotone separation and basis properties" ,i=1,2,3.In his paper,he discovered that mi-spaces have a strong connection with monotone propertry,a natural analisis of monotone normality.As we all know,network is Base’s generalization and we can get many useful topology spaces with it.So,in this paper,we generalize the space of m_I to get the relation of σ₁-spaces and weaker nested T₂ .In chapter 1,we generalize m₁-space from the opinion of network and define σ₁-spaces,namely,the spaces having a closure preserving local network at each point.We do our works as follows.In section 1.1-1.2, we discuss some necessary condition or sufficial condition of σ₁-spaces and prove that each σ₁-spaces have a function which is weaker nested T₂ . In section 1.3,we give some properties on σ₁-space and prove that every closed subspace of σ₁-space is σ₁-space,更多还原