  The Hofmann-Mislove theorem states that in a sober space, the nonempty Scott
open filters of its open set lattice correspond bijectively to its compacts
saturated sets. In this paper, the concept of $c$-well-filtered spaces is
introduced. We show that a retract of a $c$-well-filtered space is
$c$-well-filtered and a locally Lindel\"{o}f and $c$-well-filtered $P$-space is
countably sober. In particular, we obtain a Hofmann-Mislove theorem for
$c$-well-filtered spaces.
