The category of finitary biframes as the category of pointfree bispaces

Abstract

The theory of finitary biframes as order-theoretical duals of bitopological spaces is explored. The category of finitary biframes is a coreflective subcategory of that of biframes. Some of the advantages of adopting finitary biframes as a pointfree notion of bispaces are studied. In particular, it is shown that for every finitary biframe there is a biframe which plays a role analogue to that of the assembly in the theory of frames: for every finitary biframe L there is a finitary biframe A ( L ) with a universal property analogous to that of the assembly of a frame; and such that its main component is isomorphic to the ordered collection of finitary quotients of L (i.e. its pointfree bispaces). Furthermore, in the finitary biframe duality the bispace associated with A ( L ) is a natural bitopological analogue of the Skula space of the bispace associated with L . The finitary biframe duality gives us a notion of bisobriety which is weaker than pairwise Hausdorffness, incomparable with the pairwise T 1 axiom, and stronger than the pairwise T 0 axiom. The notion of pairwise T D bispaces is introduced, as a natural point-set generalization of the classical T D axiom. It is shown that in the finitary biframe duality this axiom plays a role analogous to that of the classical T D axiom for the frame duality.