Let L be a topological group acting on a locally compact Hausdorff space M as a transformation group. Let m be in M. A subset Q of M is called the local closure of the orbit Lm if Q is the smallest locally compact invariant subset of M with
m
∈
Q
m\, \in \,Q
. A partition
\[
M
=
⋃
λ
∈
∧
Q
λ
,
Q
λ
∩
Q
μ
=
∅
(
λ
≠
μ
)
M = \,\bigcup \limits _{\lambda \in \wedge } \,{Q_\lambda },\,\,\,\,{Q_{\lambda \,}}\, \cap \,\,{Q_\mu } = \,\emptyset \,\,\,\,\left ( {\lambda \ne \mu } \right )
\]
is called an LC-partition of M with respect to the L action if each
Q
λ
{Q_\lambda }
is the local closure of Lm for any m in
Q
λ
{Q_\lambda }
. Theorem. Let G be a connected Lie group, and let A and B be subgroups of G with only finitely many connected components. Suppose that B is closed. Then the factor space
G
/
B
G/B
has an LC-partition with respect to the A action.