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# Hofmann-Mislove theorem for approach spaces

The Hofmann-Mislove theorem says that compact saturated sets of a sobertopological space correspond bijectively to open filters of its open setlattice. This paper concerns an analogy of this result for approach spaces. Tothis end, the notion of compact functions of approach spaces is introduced.Such functions are an analogue of compact subsets in the enriched context. Itis shown that for a sober approach space, the inhabited and saturated compactfunctions correspond bijectively to the proper open $[0,\infty]$-filters of themetric space of its upper regular functions, which is an analogy of theHofmann-Mislove theorem for approach spaces.

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