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**1 - 2**of**2**### A Hofmann-Mislove theorem for bitopological spaces

, 2008

"... 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 ar ..."

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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.

### Representation theorems for connected compact Hausdorff spaces

, 2008

"... We present two theorems which can be used to represent compact connected Hausdorff spaces in an algebraic context, using a Stone-like representation. The first theorem stems from the work of Wallman and shows that every distributive disjunctive normal lattice is the lattice of closed sets in a uniqu ..."

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We present two theorems which can be used to represent compact connected Hausdorff spaces in an algebraic context, using a Stone-like representation. The first theorem stems from the work of Wallman and shows that every distributive disjunctive normal lattice is the lattice of closed sets in a unique up to homeomorphism connected compact Hausdorff space. The second theorem stems from the work of Jung and Sünderhauf. Introducing the notion of strong proximity involution lattices, it shows that every such lattice can be uniquely represented as the lattice of pairs of compact and open sets of connected compact Hausdorff space. As a consequence we easily obtain a somewhat surprising theorem birepresenting distributive disjunctive normal lattices and strong proximity involution lattices. 1