Embedding of Topological Posets in Hyperspaces

G. Beer ; E. A. Ok

Journal of convex analysis, Tome 30 (2023) no. 2, pp. 515-54

Résumé

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.

Détail

Classification : 06A06, 22A26, 54B20, 06F15, 06F20, 54E35, 54D45
Mots-clés : Topological poset, hyperspace, Fell topology, topological semilattice, topological po-group, topological order-embedding, radially convex metric, complete semilattice homomorphism