Topological prime radical of a group

In this paper, we consider two approaches for the definition of a topological prime radical of a topological group. In the first approach, the prime quasi-radical $\eta(G)$ is defined as the intersection of all closed prime normal subgroups of a topological group $G$. Its properties are investigated. In the second approach, we consider the set $\eta'(G)$ of all topologically strictly Engel elements of a topological group $G$. Its properties are investigated. It is proved that $\eta'(G)$ is a radical in the class of all topological groups possessing a basis of neighborhoods of the identity element consisting of normal subgroups.