Embedding of Topological Posets in Hyperspaces

We study the problem of topologically order-embedding a given topological poset

in the space of all closed subsets of

X X

which is topologized by the Fell topology and ordered by set inclusion. We show that this can be achieved whenever

(X,\preceq ) ( X , ⪯ )

is a topological semilattice (resp. lattice) or a topological po-group, and

X 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.