On rings of real valued clopen continuous functions

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)

On e-spaces and rings of real valued e-continuous functions

Whenever the closure of an open set is also open, it is called e-open and if a space have a base consisting of e-open sets, it is called e-space. In this paper we first introduce and study e-spaces and e-continuous functions (we call a function f from a space X to a space Y an e-continuous at x ∈ X if for each open set V containing f(x) there is an e-open set containing x with f ( U ) ⊆ V ). We observe that the quasicomponent of each point in a space X is determined by e-continuous functions on X and it is characterized as the largest set containing the point on which every e-continuous function on X is constant. Next, we study the rings Ce( X ) of all real valued e-continuous functions on a space X. It turns out that Ce( X ) coincides with the ring of real valued clopen continuous functions on X which is a C(Y) for a zero-dimensional space Y whose elements are the quasicomponents of X. Using this fact we characterize the real maximal ideals of Ce( X ) and also give a natural representation of its maximal ideals. Finally we have shown that Ce( X ) determines the topology of X if and only if it is a zero-dimensional space