Proposition 14

Let G be a topological group and let H be a subgroup of G. Then the homogeneous space $G/H$ is discrete if and only if H is open in G.

For the inverse images in G of the points of $G/H$ under the canonical mapping are the cosets $xH$ ($x \in G$); and these sets are open in G if and only if H is open in G.

Let X be a topological space on which a topological group G operates continuously and transitively; X is then (in the algebraic sense) a homogeneous space of G. Let x be a point of X, H_x its stabilizer. The continuous surjection s → s.x of G onto X factorizes canonically as

where f_x is the canonical mapping of G onto the homogeneous space G/H_x, and g_x is the bijection s.H_x → s.x of G/H_x onto X; moreover (Chapter I, § 3, no. 4, Proposition 6) g_x is a continuous mapping. But g_x is not necessarily a homeomorphism of G/H_x onto X (Exercise 29). If g_x is a homeomorphism for each x ∈ X, then we say that X is a topological homogeneous space (of the topological group G); for this to be so it is necessary and sufficient that, for each x ∈ E, the mapping s → s.x of G into X should be open.