A Hofmann-Mislove theorem for bitopological spaces
A `frame' is a complete lattice in which finite meets distribute over arbitrary joins. A frame homomorphism preserves finite meets and arbitrary joins leading to the category Frm. There is a dual adjunction between Top and Frm. With the duality between topological spaces and frames the authors have presented a Stone duality for bitopological spaces. In the process, the concept of \(d\)-frame has been introduced; with necessary accessories, the resulting category is denoted by \(d\)Frm; for a bitopological space the authors have taken two topologies \(\tau_+\) and \(\tau_-\) with no connection between them; morphisms between bitopological spaces are required to be continuous with respect to each of the topologies and finally, with all these, the category biTop is generated; the authors have proved a dual adjunction between biTop and \(d\)Frm. Under the additional assumption of regularity, the authors have proved a characterization theorem for subsets of a bisober space that are compact in one and closed in the other topology. This is an analogue to the celebrated Hofmann-Mislove theorem for sober spaces.
scientific article; zbMATH DE number 5318668
scientific article