Some generalizations and unifications of $C_{K}(X)$, $C_{\psi}(X)$ and $C_{\infty}(X)$

TL;DR

This paper generalizes and unifies various function rings like $C_K(X)$, $C_ ext{psi}(X)$, and $C_ ext{infinity}(X)$ using filter bases, exploring their algebraic and topological properties.

Contribution

It introduces generalized function rings based on open filter bases, extending classical theorems and characterizing their ideal and ring properties in relation to the underlying space.

Findings

$C_{ ext{infinity} ext{P}}(X)$ may not be an ideal of $C(X)$

Conditions for $C_{ ext{infinity} ext{P}}(X)$ to be an ideal are established

Characterizations of when these rings are regular or $z$-ideals

Abstract

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.