Аннотация:Suppose that G is a topological group and H is a closed subgroup of G. G/H stands
for the quotient space of G which consists of left cosets xH, where x ∈ G. We call the
space G/H a coset space. It is well known that a coset space is always homogeneous
and Tikhonov. In this paper, the author introduces P-algebraically structured maps,
a(a1)-maps and c1-maps to investigate coset spaces. The author obtains some interesting
results: (1) Suppose that X is a Lindel¨of Σ-space. Then the following are equivalent:
(a) there exists an a-mapping of X onto a space with a countable network; (b) there
exists a topological group G which contains X as a closed Gδ-subspace; (c) there exists
a topological group G which contains X as a closed subspace and satisfies the following
condition: M = {x ∈ X : there exists a Gδ-subset P of G such that x ∈ P ⊂ X} is dense
in X. (2) Suppose that a nonempty Lindel¨of Σ-subspace Y of a space X is a Gδ-subset
of the free topological group F(X) of X. Then X and F(X) are submetrizable, and
Y has a countable network. Many questions on coset spaces are posed in the paper.
These questions will give some directions for the future study of coset spaces. There are
also many old unsolved questions on homogeneous spaces that are asked again in the
paper.