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Journal of Convex Analysis 30 (2023), No. 2, 515--540
Copyright Heldermann Verlag 2023

Embedding of Topological Posets in Hyperspaces

Gerald Beer
Dept. of Mathematics, California State University, Los Angeles, U.S.A.

Efe A. Ok
Dept. of Economics and Courant Inst. of Math. Sciences, New York University, U.S.A.


We study the problem of topologically order-embedding 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 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.

Keywords: Topological poset, hyperspace, Fell topology, topological semilattice, topological po-group, topological order-embedding, radially convex metric, complete semilattice homomorphism.

MSC: 06A06, 22A26, 54B20; 06F15, 06F20, 54E35, 54D45.

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