A bitopological space, a set carrying two topologies at once, is the same thing as a topological space valued in the four-element Boolean algebra, and the two categories are isomorphic. That identification lets the specialization preorder be replaced by a Boolean-valued specialization order, from which R0, T0, T1, R1 and Hausdorff are defined uniformly, together with regularity, normality and a Urysohn lemma valued in the bitopological unit square. The resulting T1 sits strictly between join T1 and componentwise T1, and the resulting Hausdorff axiom is incomparable with both pairwise Hausdorff and order-separatedness. On the sobriety side the same identification compares d-sobriety, coming from the adjunction with d-frames, against sobriety coming from the adjunction with a slice category of frames: both are characterized by pairs of irreducible closed sets, and the second is strictly stronger. A Hofmann-Mislove theorem follows, matching inhabited saturated compact Boolean-valued sets with Scott open filters of the open-set lattice using only the slice-category structure.
A bitopological space is a set carrying two topologies at once, written (X, tau[tt], tau[ff]); Kelly introduced them in 1963 because quasi-metric and quasi-uniform structures naturally produce a pair…