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

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.