Let $\cal{ P}$ be an open filter base for a filter $\cal{ F}$ on $X$. We denote by $C^{ \cal{ P}}(X)$ ($C_{ \infty\cal{ P}}(X)$) the set of all functions $f\in C(X)$ where $Z(f)$ ($\{ x: |f(x)|< \frac{ 1}{ n}\})$ contains an element of $\cal{ P}$. First, we observe that every proper subrings in the sense of Acharyya and Ghosh (Topology Proc. 2010) has such form and vice versa. After wards, we generalize some well known theorems about $C_{ K}(X), C_{ \psi}(X)$ and $C_{ \infty}(X)$ for $C^{ \cal{ P}}(X)$ and $C_{ \infty\cal{ P}}(X)$. We observe that $C_{ \infty\cal{ P}}(X)$ may not be an ideal of $C(X)$. It is shown that $C_{ \infty\cal{ P}}(X)$ is an ideal of $C(X)$ and for each $F\in\cal{ F}$, $X\setminus \overline{ F}$ is bounded \ifif the set of non-cluster points of the filter $\cal{ F}$ is bounded. By this result, we investigate topological spaces for which $C_{ \infty\cal{ P}}(X)$ is an ideal of $C(X)$ whenever $\cal{ P}$=$\{ A\subsetneq X$: $A$ is open and $X\setminus A$ is bounded $\}$ (resp., $\cal{ P}$=$\{ A\subsetneq X$: $X\setminus A$ is finite $\}$). Moreover, we prove that $C^{ \cal{ P}}(X)$ is an essential (resp., free) ideal \ifif the set $\{ V:$ $V$ is open and $X\setminus V\in\mathcal{ F}\}$ is a $\pi$-base for $X$ (resp., $\mathcal{ F}$ has no cluster point). Finally, the filter $\cal{ F}$ for which $C_{ \infty\cal{ P}}(X)$ is a regular ring (resp., $z$-ideal) is characterized.
Categories: math.GN