The assembly of a pointfree bispace and its two variations

Abstract

We explore a pointfree theory of bitopological spaces (that is, sets equipped with two topologies). In particular, here we regard finitary biframes as duals of bitopological spaces. In particular for a finitary biframe $\mathcal {L}$ the ordered collection of all its pointfree bisubspaces (i.e. its biquotients) is studied. It is shown that this collection is bitopological in three meaningful ways. In particular it is shown that, apart from the assembly of a finitary biframe $\mathcal {L}$, there are two other structures $\mathsf {A}_{cf}(\mathcal {L})$ and $\mathsf {A}_{\pm }(\mathcal {L})$, which both have the same main component as $\mathsf {A}(\mathcal {L})$. The main component of both $\mathsf {A}_{cf}(\mathcal {L})$ and $\mathsf {A}_{\pm }(\mathcal {L})$ is the ordered collection of all biquotients of $\mathcal {L}$. The structure $\mathsf {A}_{cf}(\mathcal {L})$ being a biframe shows that the collection of all biquotients is generated by the frame of the patch-closed biquotients together with that of the patch-fitted ones. The structure $\mathsf {A}_{\pm }(\mathcal {L})$ being a biframe shows the collection of all biquotients is generated by the frame of the positive biquotients together with that of the negative ones. Notions of fitness and subfitness for finitary biframes are introduced, and it is shown that the analogues of two characterization theorems for these axioms hold. A spatial, bitopological version of these theorems is proven, in which finitary biframes whose spectrum is pairwise $T_1$ are characterized, among other things in terms of the spectrum $\mathsf {bpt}(\mathsf {A}_{cf}(\mathcal {L}))$.