Embeddings of Hausdorff semitopological semilattices in hyperspaces
UDC 515.12, 512.56 Beer and Ok showed that a locally compact and order-connected Hausdorff topological semilattice $X$ can be embedded in the space of all closed subsets of $X$ endowed with the Fell topology and ordered by set inclusion. First, we show that this result can be generalized to the case...
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| Datum: | 2026 |
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| Hauptverfasser: | , |
| Format: | Artikel |
| Sprache: | Englisch |
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Institute of Mathematics, NAS of Ukraine
2026
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| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/9085 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1866663701681012736 |
|---|---|
| author | Chu, Xiangping Li, Qingguo Chu, Xiangping Li, Qingguo |
| author_facet | Chu, Xiangping Li, Qingguo Chu, Xiangping Li, Qingguo |
| author_institution_txt_mv | [
{
"author": "Xiangping Chu",
"institution": "School of Mathematical Sciences, Anhui University, Anhui, China"
},
{
"author": "Qingguo Li",
"institution": "School of Mathematics, Hunan University, Hunan, China"
}
] |
| author_sort | Chu, Xiangping |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2026-05-30T12:44:17Z |
| description | UDC 515.12, 512.56
Beer and Ok showed that a locally compact and order-connected Hausdorff topological semilattice $X$ can be embedded in the space of all closed subsets of $X$ endowed with the Fell topology and ordered by set inclusion. First, we show that this result can be generalized to the case of Hausdorff semitopological semilattices. Second, we also prove that a locally compact Hausdorff semitopological semilattice is a topological poset. Third, we conclude that a mapping defined by $x\rightarrow{\downarrow}x$ from a locally compact lower semiclosed space $(X, \tau, \leq)$ to the space of all closed subsets of $X$ endowed with the Fell topology and ordered by set inclusion is continuous if and only if $X$ is upper open and $\leq $ is closed in $X\times X.$ Finally, we introduce the concept of $H$-closedness for a $T_{0}$ topological space. |
| doi_str_mv | 10.3842/umzh.v78i5-6.9085 |
| first_indexed | 2026-05-30T01:00:41Z |
| format | Article |
| fulltext | |
| id | umjimathkievua-article-9085 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-05-31T01:00:29Z |
| publishDate | 2026 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | |
| spelling | umjimathkievua-article-90852026-05-30T12:44:17Z Embeddings of Hausdorff semitopological semilattices in hyperspaces Embeddings of Hausdorff semitopological semilattices in hyperspaces Chu, Xiangping Li, Qingguo Chu, Xiangping Li, Qingguo Hausdorff semitopological semilattice, Embedding, Hyperspace, $H$-closed; UDC 515.12, 512.56 Beer and Ok showed that a locally compact and order-connected Hausdorff topological semilattice $X$ can be embedded in the space of all closed subsets of $X$ endowed with the Fell topology and ordered by set inclusion. First, we show that this result can be generalized to the case of Hausdorff semitopological semilattices. Second, we also prove that a locally compact Hausdorff semitopological semilattice is a topological poset. Third, we conclude that a mapping defined by $x\rightarrow{\downarrow}x$ from a locally compact lower semiclosed space $(X, \tau, \leq)$ to the space of all closed subsets of $X$ endowed with the Fell topology and ordered by set inclusion is continuous if and only if $X$ is upper open and $\leq $ is closed in $X\times X.$ Finally, we introduce the concept of $H$-closedness for a $T_{0}$ topological space. УДК 515.12, 512.56 Вкладення гаусдорфових напівтопологічних напівґраток у гіперпростори Бір та Ок встановили, що локально компактна та порядково зв'язна гаусдорфова топологічна напівґратка $X$ може бути вкладена у простір усіх замкнених підмножин $X,$ наділений топологією Фелла та впорядкований за включенням множин. По-перше, встановлено, що цей результат можна узагальнити на гаусдорфові напівтопологічні напівґратки. По-друге, доведено, що локально компактна гаусдорфова напівтопологічна напівґратка є топологічною частково впорядкованою множиною. По-третє, показано, що відображення $x\rightarrow{\downarrow}x$ з локально компактного знизу напівзамкненого простору $(X, \tau, \leq)$ у простір усіх замкнених підмножин $X,$ що наділений топологією Фелла та впорядкований за включенням множин, є неперервним тоді та лише тоді, коли $X$ є відкритим зверху, а відношення $\leq$ є замкненим у $X\times X.$ Насамкінець, введено поняття $H$-замкненості для $T_{0}$-топологічного простору. Institute of Mathematics, NAS of Ukraine 2026-05-29 Article Article https://umj.imath.kiev.ua/index.php/umj/article/view/9085 10.3842/umzh.v78i5-6.9085 Ukrains’kyi Matematychnyi Zhurnal; Vol. 78 No. 5-6 (2026); 370–371 Український математичний журнал; Том 78 № 5-6 (2026); 370–371 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/9085/10658 Copyright (c) 2026 Xiangping Chu, Qingguo Li |
| spellingShingle | Chu, Xiangping Li, Qingguo Chu, Xiangping Li, Qingguo Embeddings of Hausdorff semitopological semilattices in hyperspaces |
| title | Embeddings of Hausdorff semitopological semilattices in hyperspaces |
| title_alt | Embeddings of Hausdorff semitopological semilattices in hyperspaces |
| title_full | Embeddings of Hausdorff semitopological semilattices in hyperspaces |
| title_fullStr | Embeddings of Hausdorff semitopological semilattices in hyperspaces |
| title_full_unstemmed | Embeddings of Hausdorff semitopological semilattices in hyperspaces |
| title_short | Embeddings of Hausdorff semitopological semilattices in hyperspaces |
| title_sort | embeddings of hausdorff semitopological semilattices in hyperspaces |
| topic_facet | Hausdorff semitopological semilattice Embedding Hyperspace $H$-closed; |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/9085 |
| work_keys_str_mv | AT chuxiangping embeddingsofhausdorffsemitopologicalsemilatticesinhyperspaces AT liqingguo embeddingsofhausdorffsemitopologicalsemilatticesinhyperspaces AT chuxiangping embeddingsofhausdorffsemitopologicalsemilatticesinhyperspaces AT liqingguo embeddingsofhausdorffsemitopologicalsemilatticesinhyperspaces |