On $z$-filters and $coz$-ultrafilters

Abstract

In this article we introduce the concepts of minimal prime $z$-filter, essential $z$-filter and $r$-filter. We investigate and study the behavior of  minimal prime $z$-filters and compare them with minimal prime ideals and $coz$-ultrafilters. We show that $X$ is a $P$-space if and only if every fixed prime $z$-filter is  minimal prime. It is observed that if $X$ is a $\partial$-space and $f\in C(X)$ then $Z(f)$ is a regular closed set  if and only if  $Z[M_f]$ is an $r$-filter. The collection of all minimal prime $z$-filters will be topologized and it is proved that the space of minimal prime $z$-filters is homeomorphic to the space of  $coz$-ultrafilters. Finally, concerning to Section 3 in [{\ref{R:BD}}] it is obtained several properties and relations  between the space of minimal prime $z$-filters  and the  space of minimal prime ideals in $C(X)$.