On rings of real valued clopen continuous functions

Abstract

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