σ₁-Space And Weaker Nested T₂

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,...