On chains in $H$-closed topological pospaces

TL;DR

This paper investigates the properties of chains within $H$-closed topological pospaces, providing conditions for maximal chains to contain extremal elements, and explores the $H$-closedness of topological semilattices and chains.

Contribution

It offers new sufficient conditions for maximal chains to contain extremal elements and characterizes $H$-closed topological semilattices and chains.

Findings

Maximal chains in $H$-closed spaces can contain extremal elements under certain conditions.

Any $H$-closed topological semilattice contains a zero element.

Linearly ordered $H$-closed topological semilattices are $H$-closed topological pospaces.

Abstract

We study chains in an $H$-closed topological partially ordered space. We give sufficient conditions for a maximal chain $L$ in an $H$-closed topological partially ordered space such that $L$ contains a maximal (minimal) element. Also we give sufficient conditions for a linearly ordered topological partially ordered space to be $H$-closed. We prove that any $H$-closed topological semilattice contains a zero. We show that a linearly ordered $H$-closed topological semilattice is an $H$-closed topological pospace and show that in the general case this is not true. We construct an example an $H$-closed topological pospace with a non-$H$-closed maximal chain and give sufficient conditions that a maximal chain of an $H$-closed topological pospace is an $H$-closed topological pospace.