  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 $[0,\infty]$-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.
