弱空间式Locale

Weakly Spatial Locales

空间式locale范畴SLoc是locale范畴Loc的余反射满子范畴,但对locale乘积不封闭．本文引入弱空间式locale,证明弱空间式locale范畴WSloc为范畴Loc的余反射满子范畴,且对locale秉积封闭．还证明了一个locale A是空间式的当且仅当它的枝映射localeN(A)是弱空间式的;一个空问式locale的每一个子locale都是空间式的当且仅当它的每一个子locale是弱空间式的．最后,证明了弱空间式性在定向函子下保持不变．

The category SLoc of spatial locales is a coreflective subcategory of the category Loc of locales,but the product of spatial locales is not closed under localic product.In this note the concept of wealdy spatial locale is presented,we show that the category WSLoc of weakly spatial locales is a coreflection 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 under directed functors.