SOME GENERALIZATIONS AND UNIFICATIONS OF CK(X), Cψ(X) AND C∞(X)

Let P be an open filter base for a filter F on X. We denote by C-P (X) (C infinity P (X)) the set of all functions f is an element of C(X) where Z(f ) ({x : |f (x)| < 1/n}, for all n is an element of 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-psi (X) and C-infinity(X) for C-P (X) and C infinity P (X). We observe that C infinity P (X) may not be an ideal of C(X). It is shown that C infinity P (X) is an ideal of C(X) and for each F is an element of F , X \ <(F)over bar> 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 infinity P (X) is an ideal (of) C(X) whenever P={A subset of X: A is open and X subset of A is bounded } (resp., P={A subset of 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 is an element of F } is a p-base for X (resp., F has no cluster point). Finally, the filter F for which C infinity P (X) is a regular ring (resp., z-ideal) is characterized.