Remoteness in the category of locales

Abstract

This thesis is concerned with the study of remote sublocales which are a pointfree version of remote subsets which we de ne using van Mill's de nition of a remote collection. Unlike in van Mill's case though, the remote sublocales and remote subsets in this thesis do not necessarily need to be closed and no separation axioms are imposed on locales and spaces. We characterize both of these concepts and further show that in a T1-space, the collection of isolated points is the largest remote subset of the space. Using the motivation that remote points were initially introduced with respect to a dense subspace of the Stone- Cech compacti cation, we introduce and study properties of some versions of remote sublocales called remote (resp. remote) from a dense sublocale. We also examine localic maps that preserve and re ect remote sublocales and their versions. We prove that the localic maps whose image functions send remote sublocales to remote sublocales are precisely those with weakly open left adjoints. We also use the result about the re ection of remote sublocales to prove that the Booleanization of a locale is the largest remote sublocale of the locale, a result with no pointset topological counterpart. For the preservation and re ection of sublocales that are remote from dense sublocales, we use the Stone extension, realcompact re ector and the Lindel of re ector as particular cases. Veksler de ned a maximal nowhere dense subset of a Tychono space as a closed nowhere dense subset which is not a nowhere dense subset of any closed nowhere dense subset of the space; it is called homogeneous maximal nowhere dense in case all of its regular-closed subsets are maximal nowhere dense in the space. We introduce pointfree versions of (homogeneous) maximal nowhere dense subsets and examine a relationship between the introduced sublocales and remote sublocales where we show, among other results, that every closed nowhere dense sublocale which is remote from its supplement is maximal nowhere dense. Regarding preservation and re ection of (homogeneous) maximal nowhere dense sublocales, we show that every open localic map that sends dense elements to dense elements preserves and re ects maximal nowhere dense sublocales, and if such a localic map is further injective, then it sends homogeneous maximal nowhere dense sublocales back and forth. In the category of bilocales, we provide a comprehensive study of (i; j)-nowhere dense sublocales and subsequently introduce (i; j)-remote sublocales and prove that the (i; j)-remote sublocales of a bilocale whose i-part coincides with the total part of the bilocale are precisely the sublocales that are (i; j)-remote from dense subbilocales. For a bilocale (L; L1; L2), we introduce and study the sublocale RemBL which is the collection of all elements of L inducing the closed (i; j)-remote sublocales of L.