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

TL;DR

This paper introduces and characterizes (i,j)-Baire bilocales, exploring their properties, relationships with bispaces, and extensions to topobilocales, providing new insights into their internal structure and classifications.

Contribution

It defines (i,j)-Baire bilocales, establishes their properties, and links them to bispaces and topobilocales, offering a comprehensive framework for their analysis.

Findings

(i,j)-Baire bilocales are conservative in bilocales.

In Noetherian bilocales, (i,j)-Baireness coincides with that of the ideal bilocale.

Relative (i,j)-Baire properties depend on Booleanization subbilocales.

Abstract

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 (\tau_{i},\tau_{j})-Baireness in the category of topobilocales.