  Employing a formal analogy between ordered sets and topological spaces, over
the past years we have investigated a notion of cocompleteness for topological,
approach and other kind of spaces. In this new context, the down-set monad
becomes the filter monad, cocomplete ordered set translates to continuous
lattice, distributivity means disconnectedness, and so on. Curiously, the
dual(?) notion of completeness does not behave as the mirror image of the one
of cocompleteness; and in this paper we have a closer look at complete spaces.
In particular, we construct the "up-set monad" on representable spaces (in the
sense of L. Nachbin for topological spaces, respectively C. Hermida for
multicategories); we show that this monad is of Kock-Z\"oberlein type; we
introduce and study a notion of weighted limit similar to the classical notion
for enriched categories; and we describe the Kleisli category of our "up-set
monad". We emphasize that these generic categorical notions and results can be
indeed connected to more "classical" topology: for topological spaces, the
"up-set monad" becomes the upper Vietoris monad, and the statement "$X$ is
totally cocomplete if and only if $X^\mathrm{op}$ is totally complete"
specialises to O. Wyler's characterisation of the algebras of the Vietoris
monad on compact Hausdorff spaces.
