Consider a Lie group G (or more generally, a topological group)
acts continuously on a manifold M (or more generally, a locally
compact Hausdorff space). The quotient space X : = G \ M
is locally compact, but not always Hausdorff. In this talk, we
a method to understand the topology on such a non-Hausdorff space X.
More precisely, for a given locally compact (not necessarily Hausdorff)
space X, we construct a locally compact Hausdorff space Y,
and a map τ : X → 2Y. Then, the pair (Y,τ) has a
complete information on the topology on X. In particular,
(Y,τ) describes convergence of sequences or filters on X.