The notion of this paper is to introduce a new class of closed sets called $\eta^{\ast \ast}$-closed sets and $\eta^{\ast \ast}$ open sets in topological spaces(TS) and we studied few of its basic properties. Also, examined the relationship of $\eta^{\ast \ast}$-closed set with other sets in the TS.

Keywords: $\eta^{\ast \ast}$-closed set, $\eta^{\ast \ast}$-open set.