Revisiting Stone duality for bitopological spaces: II. sequent theories, 2-sublocales, and (simple) biframes

What a summer! Yuxu Chen, Hui Kou and Zhenchao Lyu cracked three open problems in domain theory, and I have talked about one in the July 2026 post. But let us change subjects for a while, and let us go back to possible Stone dualities for bitopological spaces. I had already mentioned d-frames a few times. I have explained why we might possibly want more in the June 2026 post, and we will pursue this, relying on a new reading of the notion of sieves from the May 2016 post, which I will call sequent theories, and which will guide us through our exploration. Fixing the notion of 2-frames of the June 2026 post, we will obtain (what I am almost certain are) Anna Laura Suarez’ finitary biframes, and a final modification will lead us to something that I will call simple biframes, which will turn out to be essentially the same thing as Banaschewski, Brümmer and Hardie’s biframes. Read the full post.