Geodesic
A Hofmann-Mislove theorem for approach spaces
Theory and applications of categories, Tome 39 (2023), pp. 21-50.

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The Hofmann-Mislove theorem says that the ordered set of open filters of the open-set lattice of a sober topological space is isomorphic to the ordered set of compact saturated sets (ordered by reverse inclusion) of that space. This paper concerns a metric analogy of this result. To this end, the notion of compact functions of approach spaces is introduced. Such functions are an analog of compact subsets in the enriched context. It is shown that for a sober approach space X, the metric space of proper open [0,∞]-filters of the metric space of upper regular functions of X is isomorphic to the opposite of the metric space of inhabited and saturated compact functions of X, establishing a Hofmann-Mislove theorem for approach spaces.
Publié le :
Classification : 18B35, 18F60, 54A05, 54B30
Keywords: Approach space, compact function, sober approach space, metric space, open [0, ∞]-filter of a metric space 
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     author = {Junche Yu and Dexue Zhang},
     title = {A {Hofmann-Mislove} theorem for approach spaces},
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Junche Yu; Dexue Zhang. A Hofmann-Mislove theorem for approach spaces. Theory and applications of categories, Tome 39 (2023), pp. 21-50. http://geodesic.mathdoc.fr/item/TAC_2023_39_a1/