On rings of real valued clopen continuous functions

Applied General Topology

Susan Afrooz¹  Masoomeh Etebar²  Fariborz Azarpanah² 

[1] Khoramshahr University of Marine Science and Technology;Shahid Chamran University of Ahvaz

关键词: 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;   

DOI  :  10.4995/agt.2018.7667

学科分类：几何与拓扑

来源: Open Journal Systems

【 摘 要 】

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 C s (X) of all real valued clopen continuous functions on a topological space X.  It is shown that every ƒ ∈ C s (X) is constant on each quasi-component in X and using this fact we show that C s (X) ≅ C(Y), where Y is a zero-dimensional s-quotient space of X.  Whenever X is locally connected, we observe  that C s (X) ≅ C(Y),  where Y is a discrete space.  Maximal ideals of C s (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 C s (X)is  fixed. It is shown that the socle of C s (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 P s -spaces, almost P s -spaces, s-basically and s-extremally disconnected spaces  are  defined  and  some  algebraic  characterizations  of  them  are given via the ring C s (X).

【 授权许可】

CC BY-NC-ND