2020/10/09 by Suarez, Anna Laura
#06D22 #Category Theory (math.CT) #FOS: Mathematics
paper · doi:10.48550/arxiv.2010.04622
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 bisubspaces). 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 T1 axiom, and stronger than the pairwise T0 axiom. The notion of pairwise TD bispaces is introduced, as a natural point-set generalization of the classical TD axiom. It is shown that in the finitary biframe duality this axiom plays a role analogous to that of the classical TD axiom for the frame duality.