On the relation between completeness and H-closedness of pospaces without infinite antichains

Abstract

We study the relation between completeness and H-closedness for topological partially ordered spaces. In general, a topological partially ordered space with an infinite antichain which is even directed complete and down-directed complete, is not H-closed. On the other hand, for a topological partially ordered space without infinite antichains, we give necessary and sufficient condition to be H-closed, using directed completeness and down-directed completeness. Indeed, we prove that  {a pospace} \(X\) is H-closed if and only if each up-directed (resp. down-directed) subset has a supremum (resp. infimum) and, for each nonempty chain \(L \subseteq X\), \( \bigvee L  \in \mathrm{cl  } {{\mathop{\downarrow} }} L\) and \( \bigwedge L  \in \mathrm{cl  } {{\mathop{\uparrow} }} L\). This extends a result of Gutik, Pagon, and Repovs [GPR].