  Let $\Delta$ be a foliation on a topological manifold $X$, $Y$ be the space
of leaves, and $p: X \to Y$ be the natural projection. Endow $Y$ with the
factor topology with respect to $p$. Then the group $\mathcal{H}(X, \Delta)$ of
foliated (i.e. mapping leaves onto leaves) homeomorphisms of $X$ naturally acts
on the space of leaves $Y$, which gives a homomorphism $\psi: \mathcal{H}(X,
\Delta) \to \mathcal{H}(Y)$. We present sufficient conditions when $\psi$ is
continuous with respect to the corresponding compact open topologies. In fact
similar results hold not only for foliations but for a more general class of
partitions $\Delta$ of locally compact Hausdorff spaces $X$.
