On maximal nowhere dense sublocales

Abstract

The aim of this paper is to study some variants of nowhere dense sublocales called maximal nowhere dense and homogeneous maximal nowhere dense sublocales. These concepts were initially introduced by Veksler in classical topology. We give some general properties of these sublocales and further examine their relationship with both inaccessible sublocales and remote sublocales. It turns out that a locale has all of its non-void nowhere dense sublocales maximal nowhere dense precisely when all of its its non-void nowhere dense sublocales are inaccessible. We show that the Booleanization of a locale is inaccessible with respect to every dense and open sublocale. In connection to remote sublocales, we prove that, if the supplement of an open dense sublocale S is homogeneous maximal nowhere dense, then every $S^{\#}$-remote sublocale is *-remote from S. Every open localic map that sends dense elements to dense elements preserves and reflects maximal nowhere dense sublocales. If such a localic map is further injective, then it sends homogeneous maximal nowhere dense sublocales back and forth.