Some generalizations and unifications of CK(X), Cψ(X) and C∞(X)M

Abstract

Let P be an open lter base for a lter F on X. We denote by C^P(X) (C_∞P(X)) the set of all functions f ∈ C(X) where Z(f) ({x : |f(x)| < ¹/_n}; ∀n ∈ N) contains an element of P. First, we observe that every subring in the sense of Acharyya and Ghosh (Topology Proc. 2010) has such form and vice versa. Afterwards, we generalize some well-known theorems about C_K(X);C_ψ(X) and C_∞(X) for C^P(X) and C_∞P(X). We observe that C_∞P(X) may not be an ideal of C(X). It is shown that C_∞P(X) is an ideal of C(X) and for each F ∈ F, X \ F is bounded if and only if the set of non-cluster points of the filter F is bounded. By this result, we investigate topological spaces X for which C_∞P(X) is an ideal of C(X) whenever P={A ⊊ X: A is open and X \ A is bounded } (resp., P={A ⊊ X:X \ A is nite }). Moreover, we prove that C^P (X) is an essential (resp., free) ideal if and only if the set {V : V is open and X \ V ∈ F} is a π-base for X (resp., F has no cluster point). Finally, the lter F for which C_∞P(X) is a regular ring (resp., z-ideal) is characterized.

Keywords: Local space, bounded subset, z-ideal, regular ring, essential ideal, F-CG_δ subset