A Hofmann-Mislove theorem for Bitopological Spaces

Authors:
Achim Jung;M. Andrew Moshier
Affiliations:
School of Computer Science, The University of Birmingham, Birmingham, United Kingdom;Department of Mathematics & Computer Science, Chapman University, Orange, CA, USA
Venue:
Electronic Notes in Theoretical Computer Science (ENTCS)
Year:
2007

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Abstract

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 Escardo'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.