Research On Local Compactness And Reflection Of Locale Category

This paper mainly studies the compact regular reflection of locally compact regular locale,the single point compaction of locally compact locale and the compact normal reflection of locale.First of all,this paper puts forward the concept of “co-compact element” and constructively defines the locally compact regular locale by it,which solves the problem that the property of locally compact can not be popularize on the locale because of uncertain definition of locally compact in the topological space.For the locally compact regular locale A,it is proved that the “ ” relationship maintains the insertion,and the whole of all the regular ideals consist locale CR(A),and CR(A)is reguale compact,so that the compact regular reflection of locally compact regular locale is also be constructed.Secondly,by using the concept of "co-compact element",a single-point compaction structure of the locally compact locale is given.That is,for any non-compact but locally compact locale,adding the element {0,1},it can prove that the newly generated lattice is a compact locale,which called S(A).In particular,for the locally compact regular locale,it proves that the locale after the singlepoint compaction is compact,and the compact regular reflection is given.Therefore,for locally compact locale,using the defined locale single-point compaction,constructs a compact regualr reflection S(A),which is differents from CR(A),and the two compact regular reflections are isomorphic.Finally,the compact regular reflection of the normal locale is studied,and the normal reflection of the compact locale is given.By proving that the normal locale is regular,and the compact regular locale is normal.For any normal locale A,it can construct its compact regular reflection CR(A).On this basis,consider constructing a normal reflection.For any compact locale A,using its compact regualr reflection CR(A)and any normal locale B,it can constructe a compact normal reflection of any compcat locale.