Remoteness in the category of locales
Loading...
Authors
Nxumalo, Mbekezeli Sibahle
Issue Date
2023-05
Type
Thesis
Language
en
Keywords
Sublocale , Localic map , Compacti cation , Bilocale , Subbilocale , Remote point , Remote sublocale , Nowhere dense sublocale , Maximal nowhere dense sublocale , Dense subbilocale , (i, j)-nowhere dense sublocale , (i, j)-remote sublocale , Space Study and Square Kilometer Area , SDG 9 Industry, Innovation and Infrastructure
Alternative Title
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.