Weak Space Locale Locale Reflection, And

In this dissertation, we investigate some properties on locales. The main results are summarized as follows.1. The concept of weakly spatial locale is presented. We show that the category WSLoc of weakly spatial locales is a coreflective subcategory of the category Loc and the localic product of weakly spatial locale is still weakly spatial. Moreover, we show that a locale A is spatial if and only if the nucleus locale N(A) is weakly spatial. Every sublocale of a locale A is spatial if and only if every sublocale of A is weakly spatial. Also we prove that the weekly spatiality is preserved by directed functors.2. By introducing regular elements, completely regular elements and zero-dimensiomal elements, we give constructive descriptions of the completely regular reflection and the zero-dimensional reflection of locales in much simple form. For a locale A, which satisfies the subdivisibility property, we show that the locale R(A) of regular elements is the regular reflection of A. Furthermore, we obtain the compact regular reflection, the completely regular reflection and the compact zero-dimensional reflection of locales by the extension lemma of flat sulocale.3. For normal locales, we prove that the Banaschewski-Mulvey's compact regular reflection construction of locales is isomorphisc to the Johnstone Wallman com-pcactification of locales. We show that a subfit semi-normal locale is normal, but the converse is not true in general. Furthermore, we generalized the main result in[76].4. By applying the compact regular reflection construction of locales given in the former chapter, we show that the compact regular reflection preserves and reflects connectedness for compactifiable locales, which satisfies the subdivisibility property.5. we give a weakly form of the composition theorem of reflectors.