More on generalizations of topology of uniform convergence and $m$-topology on $C(X)$

Abstract

This paper conglomerates our findings on the space $C(X)$ of all real valued continuous functions, under different generalizations of the topology of uniform convergence and the $m$-topology. The paper begins with answering all the questions which were left open in our previous paper on the classifications of $Z$-ideals of $C(X)$ induced by the $U_I$ and the $m_I$-topologies on $C(X)$. Motivated by the definition of $m^I$-topology, another generalization of the topology of uniform convergence, called $U^I$-topology, is introduced here. Among several other results, it is established that for a convex ideal $I$, a necessary and sufficient condition for $U^I$-topology to coincide with $m^I$-topology is the boundedness of $X\setminus \bigcap Z[I]$ in $X$. As opposed to the case of the $U_I$-topologies (and $m_I$-topologies), it is proved that each $U^I$-topology (respectively, $m^I$-topology) on $C(X)$ is uniquely determined by the ideal $I$. In the last section, the denseness of the set of units of $C(X)$ in $C_U(X)$ (= $C(X)$ with the topology of uniform convergence) is shown to be equivalent to the strong zero dimensionality of the space $X$. Also, the space $X$ is a weakly P-space if and only if the set of zero divisors (including 0) in $C(X)$ is closed in $C_U(X)$. Computing the closure of $C_{\mathcal{P}}(X)$ (=$\{f\in C(X):\text{the support of }f\in \mathcal{P}\}$ where $\mathcal{P}$ denotes the ideal of closed sets in $X$) in $C_U(X)$ and $C_m(X)$ (= $C(X)$ with the $m$-topology), the results $c{l}_U{C}_{\mathcal{P}}(X)=C_{\infty }^{\mathcal{P}}(X)$ ($=\{f\in C(X):\forall n\in \mathbb{N},\{x\in X:|f(x)|\ge \frac{1}{n}\}\in \mathcal{P}\}$) and $c{l}_m{C}_{\mathcal{P}}(X)=\{f\in C(X):f.g\in C_{\infty }^{\mathcal{P}}(X)\text{ for each }g\in C(X)\}$ are achieved.