On e-spaces and rings of real valued e-continuous functions - Susan Afrooz; Fariborz Azarpanah; Nidaah Hasan Hajee (Applied General Topology, 2023)
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.