On rings of real valued clopen continuous functions

Susan Afrooz, Fariborz Azarpanah, Masoomeh Etebar

Abstract

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).


Keywords

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

Subject classification

54C40

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References

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