The Separation Of Fibrewise Locale

Fibrewise regular and fibrewise normal are two important separation properties of fibrewise topological space.At the same time the research of new concept fibrewise locale provides space to think,for fibrewise locale,whether or not some conditions are satisfied for fibrewise locale,it is fibrewise regular or fibrewise normal.In addition to the fibrewise regular and the fibrewise normal,and whether it has other separation properties.Hence this work the definition of fibrewise locale is presented on the basis of locale theory,the definition of fibrewise separation and related properties are illustrated mainly in the locale theory,and combined with the definition of fibrewise regular and fibrewise normal in fibrewise topological space,no point description and theorem of fibrewise regular and fibrewise normal are given in the space type fibrewise locale,and at the same time the definition and proposition of fibrewise regular and fibrewise normal of space type fibrewise locale as a special case of the space type fibrewise locale,and in the process of the discussion equivalent description is given with the original fibrewise topological space.The article also related to other properties of separation properties of space type fibrewise locale,such as the definition and related proposition of fibrewise T0、fibrewise R0、 fibrewise T2、fibrewise T21/(?)、 fibrewise completely regular.The main conclusions of this paper are as follows:Theorem3.1.2:let X be a fibrewise topological space over B,Ω(X)andΩ(B)are said open set lattice in X and B,then fibrewise topological space X is fiberwise regular if and only if fiberwise localeΩ(X)(?)Ω(B) is fiberwise regular.Theorem3.1.4:let LX(?)LB be fiberwise locale,f:LB�'LX be a frame homomorphism and LXi is sublocale of Lx. If the fiberwise locale LX(?)LB is the fiberwise regular, then fiberwise locale LXi (?) LB is the fiberwise regular. Theorem4.1.2:let X be a fibrewise topological space over B,Ω(X)andΩ(B)are said open set lattice in X and B,then fibrewise topological space X is fiberwise normal if and only if fiberwise locale Ω(X)(?)Ω(B) is fiberwise normal.