Embedding of Topological Posets in Hyperspaces

TL;DR

This paper investigates how topological posets can be embedded into hyperspaces of closed sets, establishing conditions for such embeddings and deriving applications like a metrization theorem and fixed point results.

Contribution

It provides new embedding theorems for topological posets into hyperspaces under specific conditions, extending classical results and applications.

Findings

Embedding of topological semilattices and lattices into hyperspaces achieved

A locally compact version of the Urysohn-Carruth metrization theorem established

New fixed point theorem of Tarski-Kantorovich type proved

Abstract

We study the problem of topologically order-embedding a given topological poset X 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 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.