Abstrakt
The frame S-c(L) generated by closed sublocales of a locale L is known to be a natural Boolean ("discrete") extension of a subfit L; also it is known to be its maximal essential extension. In this paper we first show that it is an essential extension of any L and that the maximal essential extensions of L and S-c(L) are isomorphic.
The construction S-c is not functorial; this leads to the question of individual liftings of homomorphisms L -> M to homomorphisms S-c(L) -> S-c(M). This is trivial for Boolean L and easy for a wide class of spatial L, M.
Then, we show that one can lift all h : L -> 2 for weakly Hausdorff L (and hence the spectra of L and S-c(L) are naturally isomorphic), and finally present liftings of h : L -> M for regular L and arbitrary Boolean M.