Detecting topological groups which are (locally) homeomorphic to LF-spaces

We prove that a topological group G is (locally) homeomorphic to an LF-space if G = ⋃ n ∈ ω G n for some increasing sequence of subgroups ( G n ) n ∈ ω such that(1) y neighborhoods U n ⊂ G n , n ∈ ω , of the neutral element e ∈ G n ⊂ G , the set ⋃ n = 1 ∞ U 0 U 1 ⋯ U n is a neighborhood of e in G; roup G n is (locally) homeomorphic to a Hilbert space; ery n ∈ N the quotient map G n → G n / G n − 1 is a locally trivial bundle; finitely many numbers n ∈ N each Z-point in the quotient space G n / G n − 1 = { x G n − 1 : x ∈ G n } is a strong Z-point.