Remoteness in the category of bilocales

TL;DR

This paper introduces and analyzes (i,j)-remote sublocales in bilocales, extending the concept of remoteness from locale theory to bilocales and related structures, with comprehensive theoretical results.

Contribution

It defines (i,j)-remote sublocales in bilocales, studies their properties, and extends the concept to bitopological spaces and normed lattices, providing a new framework in locale theory.

Findings

(i,j)-remote sublocales miss all (i,j)-nowhere dense sublocales.

In balanced bilocales, (i,j)-nowhere dense sublocales are characterized by their closures.

Extension of (i,j)-remoteness to bitopological spaces and normed lattices.

Abstract

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 (\tau_{i},\tau_{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,\tau_{1},\tau_{2}) is (\tau_{i},\tau_{j})-remote if and only if the induced sublocale \widetilde{A} of \tau_{1}\vee\tau_{2} is (i,j)-remote.