  We study the problem of topologically order-embedding a given topological
poset X in the space of all closed subsets of X which is topologized by the
Fell topology and ordered by set inclusion. We show that this can be achieved
whenever X is a topological semilattice (resp. lattice) or a topological
po-group, and X is locally compact and order-connected (resp. connected). We
give limiting examples to show that these results are tight, and provide
several applications of them. In particular, a locally compact version of the
Urysohn-Carruth metrization theorem is obtained, a new fixed point theorem of
Tarski-Kantorovich type is proved, and it is found that every locally compact
and connected Hausdorff topological lattice is a completely regular ordered
space.

General Topology