Compact Zero-dimensional Reflection And Nucleus Of Locale Category

Reflective structure is an important research content of Locale theory.It is the main research method to construct the ideal of specific property and give the corresponding reflection structure.In this paper,we study the different reflection structures of locale and discuss the necessary and sufficient conditions for the consistency of zero-dimensional reflection and compact zero-dimensional reflection.Using the nucleus,the compact regular reflection of normal locale and the compact zero-dimensional reflection of locale are given.Firstly,the concept of B-compact locale is introduced in locale,By using the different equivalent construction of zero-dimensional reflection and compact zero-dimensional reflection,it is proved that the zero-dimensional reflection of locale is consistent with the compact zero-dimensional reflection only if the locale is B-compact.In particular,n-dimensional Euclidian Spaces are B-compact,so the zero-dimensional reflection is consistent with the compact zero-dimensional reflection.Secondly,T₁ separation in topological space is generalized to locale.On this basis,the nucleus on the locale A is defined,and the sublocale of J-fixed ideals is denoted by Idl(A)_J,We prove that I is a regular ideal,for I?Idl(A)_J in the normal locale,finally provedIdl(A)_J is the compact regular reflection of A in locale,This conclusion is a further extension of P.T.Johnstone's conclusion on compact regular reflection of normal and regular locales.Finally,the method of constructing compact regular reflection by using nucleus is applied to the construction of compact zero-dimensional reflection.The appropriate nucleus on the locale L is given,and the corresponding Idl_k(L)is obtained,which proves that it is a compact zero dimensional locale.In the locale category,we prove the existing compact zero-dimensional reflection construction is isomorphic to the Idl_k(L),and give different construction methods of compact zero-dimensional reflection.