  As composites of constant, (co)product, identity, and powerset functors,
Kripke polynomial functors form a relevant class of $\mathsf{Set}$-functors in
the theory of coalgebras. The main goal of this paper is to expand the theory
of limits in categories of coalgebras of Kripke polynomial functors to the
context of quantale-enriched categories. To assume the role of the powerset
functor we consider "powerset-like" functors based on the Hausdorff
$\mathsf{V}$-category structure. As a starting point, we show that for a
lifting of a $\mathsf{SET}$-functor to a topological category $\mathsf{X}$ over
$\mathsf{Set}$ that commutes with the forgetful functor, the corresponding
category of coalgebras over $\mathsf{X}$ is topological over the category of
coalgebras over $\mathsf{Set}$ and, therefore, it is "as complete" but cannot
be "more complete". Secondly, based on a Cantor-like argument, we observe that
Hausdorff functors on categories of quantale-enriched categories do not admit a
terminal coalgebra. Finally, in order to overcome these "negative" results, we
combine quantale-enriched categories and topology \emph{\`a la} Nachbin.
Besides studying some basic properties of these categories, we investigate
"powerset-like" functors which simultaneously encode the classical Hausdorff
metric and Vietoris topology and show that the corresponding categories of
coalgebras of "Kripke polynomial" functors are (co)complete.
