
Journal of Convex Analysis 30 (2023), No. 2, 515540 Copyright Heldermann Verlag 2023 Embedding of Topological Posets in Hyperspaces Gerald Beer Dept. of Mathematics, California State University, Los Angeles, U.S.A. gbeer@cslanet.calstatela.edu Efe A. Ok Dept. of Economics and Courant Inst. of Math. Sciences, New York University, U.S.A. efe.ok@nyu.edu [Abstractpdf] We study the problem of topologically orderembedding a given topological poset $(X,\preceq)$ 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 $(X,\preceq )$ is a topological semilattice (resp. lattice) or a topological pogroup, and $X$ is locally compact and orderconnected (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 UrysohnCarruth metrization theorem is obtained, a new fixed point theorem of TarskiKantorovich type is proved, and it is found that every locally compact and connected Hausdorff topological lattice is a completely regular ordered space. Keywords: Topological poset, hyperspace, Fell topology, topological semilattice, topological pogroup, topological orderembedding, radially convex metric, complete semilattice homomorphism. MSC: 06A06, 22A26, 54B20; 06F15, 06F20, 54E35, 54D45. [ Fulltextpdf (211 KB)] for subscribers only. 