Local theorems for topological groups

Topological groups with various systems of closed invariant subgroups are studied, and theorems generalizing Mal'cev's local theorems for Kurosh–Chernikov class to locally compact groups are proved. As an application, information is obtained on systems of closed invariant subgroups of a locally compact, locally prosolvable or locally pronilpotent group. Problem 3.37 in the Kourovka notebook on the Frattini subgroup of a locally compact, locally pronilpotent group is also solved.
A new topologization of the set of closed subsets of a topological space is applied to prove the local theorems.
Bibliography: 9 titles.