A Hofmann-Mislove theorem for Bitopological Spaces

We present a Stone duality for bitopological spaces in analogy to the duality between topological spaces and frames, and discuss the resulting notions of sobriety and spatiality. Under the additional assumption of regularity, we prove a characterisation theorem for subsets of a bisober space that are compact in one and closed in the other topology. This is in analogy to the celebrated Hofmann-Mislove theorem for sober spaces. We link the characterisation to Taylor's and Escardó's reading of the Hofmann-Mislove theorem as continuous quantification over a subspace. As an application, we define locally compact d-frames and show that these are always spatial.