On rings of real valued clopen continuous functions

Authors

  • Susan Afrooz Khoramshahr University of Marine Science and Technology
  • Fariborz Azarpanah Shahid Chamran University of Ahvaz
  • Masoomeh Etebar Shahid Chamran University of Ahvaz

DOI:

https://doi.org/10.4995/agt.2018.7667

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

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).

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Author Biographies

Fariborz Azarpanah, Shahid Chamran University of Ahvaz

Department of Mathematics

Masoomeh Etebar, Shahid Chamran University of Ahvaz

Department of Mathematics

References

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Published

2018-10-04

How to Cite

[1]
S. Afrooz, F. Azarpanah, and M. Etebar, “On rings of real valued clopen continuous functions”, Appl. Gen. Topol., vol. 19, no. 2, pp. 203–216, Oct. 2018.

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