2024/07/20 by Mbekezeli Nxumalo, Nxumalo, Mbekezeli
Business, Management and Accounting · #06D22 #Banking Systems and Strategies #FOS: Mathematics #General Topology (math.GN) #Taxation and Legal Issues
paper · pdf · doi:10.48550/arxiv.2407.14755
openalex publication_date 2024/07/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 (τi,τj)-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-TD bitopological spaces, a subset A of a bitopological space (X,τ1,τ2) is (τi,τj)-remote if and only if the induced sublocale \widetildeA of τ1\veeτ2 is (i,j)-remote.