Abstract

We study the problem of topologically order-embedding a given topological poset (X,⪯)(X,\preceq) in the space of all closed subsets of XX which is topologized by the Fell topology and ordered by set inclusion. We show that this can be achieved whenever (X,⪯)(X,\preceq ) is a topological semilattice (resp. lattice) or a topological po-group, and XX 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.

Author information

Contact details are reproduced from the original publication and may be historical.

Efe A. Ok

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

efe.ok@nyu.edu

Keywords

  • Topological poset
  • hyperspace
  • Fell topology
  • topological semilattice
  • topological po-group
  • topological order-embedding
  • radially convex metric
  • complete semilattice homomorphism

Mathematics Subject Classification

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

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Suggested citation

G. Beer, E. A. Ok. “Embedding of Topological Posets in Hyperspaces.” Journal of Convex Analysis 30 (2023), No. 2, 515–540. https://doi.org/10.68381/jca30030

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Published by Heldermann Verlag, 2023. Rights now held by Banach Press LLC.