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...
Збережено в:
| Дата: | 2018 |
|---|---|
| Автор: | |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Lugansk National Taras Shevchenko University
2018
|
| Теми: | |
| Онлайн доступ: | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/748 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Algebra and Discrete Mathematics |
Репозитарії
Algebra and Discrete Mathematics| id |
oai:ojs.admjournal.luguniv.edu.ua:article-748 |
|---|---|
| record_format |
ojs |
| spelling |
oai:ojs.admjournal.luguniv.edu.ua:article-7482018-04-26T00:55:03Z On the relation between completeness and H-closedness of pospaces without infinite antichains Yokoyama, Tomoo H-closed, pospace, directed complete Primary 06A06, 06F30; Secondary 54F05, 54H12 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]. Lugansk National Taras Shevchenko University 2018-04-26 Article Article Peer-reviewed Article application/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/view/748 Algebra and Discrete Mathematics; Vol 15, No 2 (2013) 2415-721X 1726-3255 en https://admjournal.luguniv.edu.ua/index.php/adm/article/view/748/278 Copyright (c) 2018 Algebra and Discrete Mathematics |
| institution |
Algebra and Discrete Mathematics |
| collection |
OJS |
| language |
English |
| topic |
H-closed pospace directed complete Primary 06A06 06F30; Secondary 54F05 54H12 |
| spellingShingle |
H-closed pospace directed complete Primary 06A06 06F30; Secondary 54F05 54H12 Yokoyama, Tomoo On the relation between completeness and H-closedness of pospaces without infinite antichains |
| topic_facet |
H-closed pospace directed complete Primary 06A06 06F30; Secondary 54F05 54H12 |
| format |
Article |
| author |
Yokoyama, Tomoo |
| author_facet |
Yokoyama, Tomoo |
| author_sort |
Yokoyama, Tomoo |
| title |
On the relation between completeness and H-closedness of pospaces without infinite antichains |
| title_short |
On the relation between completeness and H-closedness of pospaces without infinite antichains |
| title_full |
On the relation between completeness and H-closedness of pospaces without infinite antichains |
| title_fullStr |
On the relation between completeness and H-closedness of pospaces without infinite antichains |
| title_full_unstemmed |
On the relation between completeness and H-closedness of pospaces without infinite antichains |
| title_sort |
on the relation between completeness and h-closedness of pospaces without infinite antichains |
| description |
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]. |
| publisher |
Lugansk National Taras Shevchenko University |
| publishDate |
2018 |
| url |
https://admjournal.luguniv.edu.ua/index.php/adm/article/view/748 |
| work_keys_str_mv |
AT yokoyamatomoo ontherelationbetweencompletenessandhclosednessofpospaceswithoutinfiniteantichains |
| first_indexed |
2024-04-12T06:25:23Z |
| last_indexed |
2024-04-12T06:25:23Z |
| _version_ |
1796109253154439168 |