The extensions of some convergence phenomena in topological groups

In this paper sequentially compact sets, weakly first-countable sets and generalized metric sets in extensions of topological groups are studied. Some three space properties on convergence phenomena are obtained. It is shown that (1) if H is a closed subgroup of a topological group G such that all sequentially compact subsets of both the group H   and the quotient space G/HG/H are sequential, then all sequentially compact subsets of G are sequential; (2) let H be a closed and second-countable subgroup of a topological group G, then G   is a topological sum of ℵ0ℵ0-subspaces if the quotient space G/HG/H is a local ℵ0ℵ0-space; (3) let H be a locally compact and metrizable subgroup of a topological group G, then G   is sequential if the quotient space G/HG/H is sequential.