On rings of real valued clopen continuous functions

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 ? ∈ 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).