ON AN OVER-RING C(X)∆ OF C(X)

Abstract

Our aim in this paper is to introduce a ring of functions defined on a topological space X having a special property. By C(X)∆ we denote the set of all realvalued functions defined on the topological space X, the discontinuity set of elements of which are members of ∆ ⊆ P(X), where ∆ satisfies the following properties: (i) for each x ∈ X, {x} ∈ ∆, (ii) for A, B ∈ P(X) with A ⊆ B, B ∈ ∆ implies that A ∈ ∆ and (iii) for A, B ∈ ∆, A ∪ B ∈ ∆. This C(X)∆ is an over-ring of C(X), moreover, C(X) ⊆ C(X)F ⊆ C(X)∆ ⊆ RX. The ring C(X)∆ is also almost regular. We study the ∆-completely separated sets and C∆-embedded subsets of X. Complete characterizations of fixed maximal ideals are then done and algebraic properties of C(X)∆ have been studied. In [6], the authors have introduced FP-spaces, for which the ring C(X)F is regular. Here we have generalized the notion of FP-spaces in the context of C(X)∆, so that the ring in question becomes regular. As a result, ∆Pspaces have been introduced, it has been proved that every P-space is a ∆P-space and examples are given in support of the fact that there exist ∆P-spaces which are not P-spaces.