Actions of groups of foliated homeomorphisms on spaces of leaves

Let $\mathrm{\Delta }$ 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 $\mathcal{H}(X,\mathrm{\Delta })$ 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)$\mathrm{.}Wepresentsufficientconditionswhen$\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 $\mathrm{\Delta }$ of locally compact Hausdorff spaces $X$.