Topology generated by cluster systems

Abstract

In this paper, we prove that $(X,\tau )$ and thenew topology $(X,{\tau }_{\mathcal{E}})$ have the same semiregularization if$\mathcal{E}$ is a $\pi$-network in $X$ with the property $\mathcal{H}$. Also, wediscuss the properties of $\mathcal{E},{\tau }_{\mathcal{E}}$ and study generalizedVolterra spaces and discuss their properties. We show that${\tau }_{\mathcal{E}}$ coincides with the $\star$-topology for a particular$\mathcal{E}$.