On the diffeomorphism groups of foliated manifolds

In this paper, we introduce a certain topology on the group $\mathrm{Diff}_F(M)$ of all $C^r$-diffeomorphisms of the foliated manifold $(M;F)$, where $r\ge0$. This topology depends on the foliation and is called the $F$-compact-open topology. It coincides with the compact-open topology when $F$ is an $n$-dimensional foliation. If the codimension of the foliation is $n$, then the convergence in this topology coincides with the pointwise convergence, where $n=\dim M$. We prove that some subgroups of the group $\mathrm{Diff}_F(M)$ are topological groups with the $F$-compact-open topology. Throughout this paper, we use smoothness of the class $C^{\infty}$.