On rings of real valued clopen continuous functions

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 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).