ON A GENERALIZED VERSION OF $m$-TOPOLOGY AND $U$-TOPOLOGY IN THE OVER-RING $C(X)_{_{\Delta}}$ OF $C(X)$

In this paper, we generalize the $m$-topology and the $U$-topology of $C(X)$ to its over-ring $C(X)_{_{\Delta}}$. The generalized versions will be referred to as the $m_{_{I}}^{^{\Delta}}$-topology and the $u_{_{I}}^{^{\Delta}}$-topology respectively. We define $A^\Delta_I=\{f\in C(X)_{_{\Delta}}: |f(x)|\leq M$, for all $x\in Z\setminus H$, for some $Z\in Z_{_\Delta}(I)$ and $H\in \Delta\}$, which turns out to be the component of $\underline 0$ in $C(X)_{_{\Delta}}$ with the $u^{^{\Delta}}_{_{I}}$-topology. Next we define $I_{\psi_{\Delta}}(X)=\{f\in C(X)_{_{\Delta}}:$ given $g\in C(X)_{_{\Delta}}$, there exists $M > 0$, $Z\in Z_{_\Delta}(I)$ and $H\in \Delta$ such that $|f(x).g(x)| \le M$ for each $x\in Z\setminus H\}$. This set will be seen to play a key role in determining the connected ideals in $C(X)_{_{\Delta}}$ with the $m_{_{I}}^{^{\Delta}}$-topology. It is observed that $C(X)_{_{\Delta}}$ with the $m_{_{I}}^{^{\Delta}}$-topology is a topological ring, whereas $C(X)_{_{\Delta}}$ with the $u_{_{I}}^{^{\Delta}}$-topology is not so. Finally, we give several necessary and sufficient conditions for the coincidence of the $u^{^{\Delta}}_{_{I}}$-topology and the $m^{^{\Delta}}_{_{I}}$-topology on $C(X)_{_{\Delta}}$.