Paracompactness And Reflective Properties Of Local Categories

The separation and compactness in the research of locale theory,and the construction of reflection on the relevant locale is always the hot issues in the discipline.In this paper,we discuss the compact regular reflection constructed by paracompact regular and completely paracompact regular locales,and the insertion problem on countable paracompact and normal locales.First of all,the normal properties of paracompact regular spaces in topology are uniformly extended to locales,and through the insertion of normal locales,the normal ideal lattice C_R(L) Itself.The constructive description of compact regular reflection of paracompact regular locale is given,that is,if localeis paracompact regular locale,then C_R(L) Is the paracompact regular reflection of.It is fully proved that the compact regular reflection of paracompact regular locale exists and can be constructed.Secondly,through the increasing cover lemma of countable paracompact,we obtain two increasing covers with waybelow relation.On the basis of these two increasing covers,we construct a piecewise function and further construct a real function h(1/n,-) Satisfying the insertion condition.We also apply the method of constructing real functions from piecewise functions to obtain normal insertivity,and combine the construction and proof methods of two insertivity to find the connection between the two insertivity,and then obtain the relationship between the corresponding separability and compactness.Finally,using the method of reflection on similar and countable paracompact locales,applying the properties of normal locales and completely regular paracompact locales,and then using the conclusion of related ideal lattices,we give the compact regular reflection of completely paracompact regular locales.Then we can get the different methods to construct the binary relationship with He Wei and get the corresponding conclusions.