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.
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
[Abstract-pdf]
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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