On rings of real valued clopen continuous functions
Among variant kinds of strong continuity in the literature, the clopen continuity or cl-supercontinuity (i.e., inverse image of every open set is a union of clopen sets) is considered in this paper. We investigate and study the ring Cs(X) of all real valued clopen continuous functions on a topologi...
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| Format: | Article |
| Language: | English |
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Universitat Politècnica de València
2018-10-01
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| Series: | Applied General Topology |
| Subjects: | |
| Online Access: | https://polipapers.upv.es/index.php/AGT/article/view/7667 |
| _version_ | 1819079539350831104 |
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| author | Susan Afrooz Fariborz Azarpanah Masoomeh Etebar |
| author_facet | Susan Afrooz Fariborz Azarpanah Masoomeh Etebar |
| author_sort | Susan Afrooz |
| collection | DOAJ |
| description | Among variant kinds of strong continuity in the literature, the clopen continuity or cl-supercontinuity (i.e., inverse image of every open set is a union of clopen sets) is considered in this paper. We investigate and study the ring Cs(X) of all real valued clopen continuous functions on a topological space X. It is shown that every ƒ ∈ Cs(X) is constant on each quasi-component in X and using this fact we show that Cs(X) ≅ C(Y), where Y is a zero-dimensional s-quotient space of X. Whenever X is locally connected, we observe that Cs(X) ≅ C(Y), where Y is a discrete space. Maximal ideals of Cs(X) are characterized in terms of quasi-components in X and it turns out that X is mildly compact(every clopen cover has a finite subcover) if and only if every maximal ideal of Cs(X)is fixed. It is shown that the socle of Cs(X) is an essential ideal if and only if the union of all open quasi-components in X is s-dense. Finally the counterparts of some familiar spaces, such as Ps-spaces, almost Ps-spaces, s-basically and s-extremally disconnected spaces are defined and some algebraic characterizations of them are given via the ring Cs(X). |
| first_indexed | 2024-12-21T19:30:36Z |
| format | Article |
| id | doaj.art-27ed458bb32a410291d032e5cf666e2f |
| institution | Directory Open Access Journal |
| issn | 1576-9402 1989-4147 |
| language | English |
| last_indexed | 2024-12-21T19:30:36Z |
| publishDate | 2018-10-01 |
| publisher | Universitat Politècnica de València |
| record_format | Article |
| series | Applied General Topology |
| spelling | doaj.art-27ed458bb32a410291d032e5cf666e2f2022-12-21T18:52:43ZengUniversitat Politècnica de ValènciaApplied General Topology1576-94021989-41472018-10-0119220321610.4995/agt.2018.76677096On rings of real valued clopen continuous functionsSusan Afrooz0Fariborz Azarpanah1Masoomeh Etebar2Khoramshahr University of Marine Science and TechnologyShahid Chamran University of AhvazShahid Chamran University of AhvazAmong variant kinds of strong continuity in the literature, the clopen continuity or cl-supercontinuity (i.e., inverse image of every open set is a union of clopen sets) is considered in this paper. We investigate and study the ring Cs(X) of all real valued clopen continuous functions on a topological space X. It is shown that every ƒ ∈ Cs(X) is constant on each quasi-component in X and using this fact we show that Cs(X) ≅ C(Y), where Y is a zero-dimensional s-quotient space of X. Whenever X is locally connected, we observe that Cs(X) ≅ C(Y), where Y is a discrete space. Maximal ideals of Cs(X) are characterized in terms of quasi-components in X and it turns out that X is mildly compact(every clopen cover has a finite subcover) if and only if every maximal ideal of Cs(X)is fixed. It is shown that the socle of Cs(X) is an essential ideal if and only if the union of all open quasi-components in X is s-dense. Finally the counterparts of some familiar spaces, such as Ps-spaces, almost Ps-spaces, s-basically and s-extremally disconnected spaces are defined and some algebraic characterizations of them are given via the ring Cs(X).https://polipapers.upv.es/index.php/AGT/article/view/7667clopen continuous (cl-supercontinuous)zero-dimensionalPs-spacealmost Ps-spaceBaer ringp.p. ringquasi-componentsoclemildly compacts-basically and s-extremally disconnected space |
| spellingShingle | Susan Afrooz Fariborz Azarpanah Masoomeh Etebar On rings of real valued clopen continuous functions Applied General Topology clopen continuous (cl-supercontinuous) zero-dimensional Ps-space almost Ps-space Baer ring p.p. ring quasi-component socle mildly compact s-basically and s-extremally disconnected space |
| title | On rings of real valued clopen continuous functions |
| title_full | On rings of real valued clopen continuous functions |
| title_fullStr | On rings of real valued clopen continuous functions |
| title_full_unstemmed | On rings of real valued clopen continuous functions |
| title_short | On rings of real valued clopen continuous functions |
| title_sort | on rings of real valued clopen continuous functions |
| topic | clopen continuous (cl-supercontinuous) zero-dimensional Ps-space almost Ps-space Baer ring p.p. ring quasi-component socle mildly compact s-basically and s-extremally disconnected space |
| url | https://polipapers.upv.es/index.php/AGT/article/view/7667 |
| work_keys_str_mv | AT susanafrooz onringsofrealvaluedclopencontinuousfunctions AT fariborzazarpanah onringsofrealvaluedclopencontinuousfunctions AT masoomehetebar onringsofrealvaluedclopencontinuousfunctions |