arXiv subjects

Mbekezeli Nxumalo

Publications and source records attributed to Mbekezeli Nxumalo.

4 recordsLinked to original sources

arXiv2024

Remoteness in the category of bilocales

In locale theory, a sublocale is said to be remote in case it misses every nowhere dense sublocale. In this paper, we introduce and study a new class of sublocales in the category of bilocales, namely (i,j)-remote sublocales. These are bilocalic counterparts of remote sublocales and are the sublocales missing every (i,j)-nowhere dense sublocale, with (i,j)-nowhere dense sublocales being bilocalic counterparts of (τ_{i},τ_{j})-nowhere dense subsets in bitopological spaces. A comprehensive study of (i,j)-nowhere dense sublocales is given and we show that in the class of balanced bilocales, a sublocale is (i,j)-nowhere dense if and only if its bilocale closure is nowhere dense. We also consider weakly (i,j)-remote sublocales which are those sublocales missing every clopen (i,j)-nowhere dense sublocale. Furthermore, we extend (i,j)-remoteness to the categories of bitopological spaces as well as normed lattices. In the class of \sup-T_{D} bitopological spaces, a subset A of a bitopological space (X,τ_{1},τ_{2}) is (τ_{i},τ_{j})-remote if and only if the induced sublocale \widetilde{A} of τ_{1}\veeτ_{2} is (i,j)-remote.

math.GN↗

arXiv2024

On the category of (i,j)-Baire Bilocales

We define and characterize the notion of (i,j)-Baireness for bilocales. We also give internal properties of (i,j)-Baire bilocales which are not translated from properties of (i,j)-Baireness in bispaces. It turns out (i,j)-Baire bilocales are conservative in bilocales, in the sense that a bitopological space is almost (i,j)-Baire if and only if the bilocale it induces is (i,j)-Baire. Furthermore, in the class of Noetherian bilocales, (i,j)-Baireness of a bilocale coincides with (i,j)-Baireness of its ideal bilocale. We also consider relative versions of (i,j)-Baire where we show that a bilocale is (i,j)-Baire only if the subbilocale induced by the Booleanization is (i,j)-Baire. We use the characterization of (i,j)-Baire bilocales to introduce and characterize (τ_{i},τ_{j})-Baireness in the category of topobilocales.

math.GN↗

arXiv2024

Some variants of remote sublocales

We introduce and study some variants of remote sublocales, namely sublocales that are remote from dense sublocales and those that are *remote from dense sublocales. We show that the coframe of sublocales coincides with the collection of all sublocales remote from the Booleanization. Furthermore, the supplement of the Booleanization of any locale is the largest sublocale *remote from the Booleanization. We give conditions on localic maps such that their induced localic image and pre-image functions preserve sublocales that are remote (resp. *remote) from dense sublocales. We introduce new types of localic maps called f-remote preserving maps.

math.GN↗

arXiv2024

On maximal nowhere dense sublocales

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.

math.GN↗