  Let $Aut(X,\mathcal{B})$ be the group of all Borel automorphisms of a
standard Borel space $(X,\mathcal{B})$. We study topological properties of
$Aut(X,\mathcal{B})$ with respect to the uniform and weak topologies, $\tau$
and $p$, defined in [Bezuglyi S., Dooley A.H., Kwiatkowski J., Topologies on
the group of Borel automorphisms of a standard Borel space, Preprint, 2003]. It
is proved that the class of smooth automorphisms is dense in $(Aut(X,\mathcal
B),p)$. Let $Ctbl(X)$ denote the group of Borel automorphisms with countable
support. It is shown that the topological group $Aut_0(X,\mathcal
B)=Aut(X,\mathcal{B})/Ctbl(X)$ is path-connected with respect to the quotient
topology $\tau_0$. It is also proved that $Aut_0(X,\mathcal B)$ has the Rokhlin
property in the quotient topology $p_0$, i.e., the action of $Aut_0(X,\mathcal
B)$ on itself by conjugation is topologically transitive.
