Embedding of Topological Posets in Hyperspaces
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.
Cite
@article{arxiv.2111.11575,
title = {Embedding of Topological Posets in Hyperspaces},
author = {Gerald Beer and Efe A. Ok},
journal= {arXiv preprint arXiv:2111.11575},
year = {2021}
}