  Let $\CR$ be the set of all bounded linear operators between Hilbert spaces
$\cH, \cK$. This paper is devoted to the study of the topological properties of
$\CR$ if certain natural metrics are considered on it. We also define an action
of the group $\G_\cH\times\G_\cK$ on $\CR$ and determine the orbits of this
action. These orbits determine a stratification of the set of Fredholm and
semi-Fredholm operators. Finally, we calculate the distance, with respect to
some of the metrics mentioned above, between different orbits of $\CR$.
