SOME GENERALIZATIONS AND UNIFICATIONS OF C_K(X), C_Ψ(X) AND C_∞(X)

摘要

Let P be an open filter base for a filter F on X. We denote by CP (X) (C∞P (X)) the set of all functions f ∈ C(X) where Z(f ) ({x : |f (x)| <}, ∀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 CK (X), Cψ (X) and C∞(X) for CP (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 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 finite }). Moreover, we prove that CP (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 filter F for which C∞P (X) is a regular ring (resp., z-ideal) is characterized.