On rings of real valued clopen continuous functions
[EN] 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 topo...
| Autores: | , , |
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| Tipo de recurso: | artículo |
| Fecha de publicación: | 2018 |
| País: | España |
| Institución: | Universitat Politècnica de València (UPV) |
| Repositorio: | RiuNet. Repositorio Institucional de la Universitat Politécnica de Valéncia |
| Idioma: | inglés |
| OAI Identifier: | oai:riunet.upv.es:10251/109446 |
| Acceso en línea: | https://riunet.upv.es/handle/10251/109446 |
| Access Level: | acceso abierto |
| Palabra clave: | 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 |
| Sumario: | [EN] 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). |
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