On the relation between completeness and H-closedness of pospaces without infinite antichains
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 partial...
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| Date: | 2018 |
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| Main Author: | |
| Format: | Article |
| Language: | English |
| Published: |
Lugansk National Taras Shevchenko University
2018
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| Subjects: | |
| Online Access: | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/748 |
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| Journal Title: | Algebra and Discrete Mathematics |
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Algebra and Discrete Mathematics| Summary: | 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]. |
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