  We study topological properties of the symmetric inverse topological
semigroup of finite transformations $\mathscr{I}_\lambda^n$ of the rank
$\leqslant n$. We show that the topological inverse semigroup
$\mathscr{I}_\lambda^n$ is algebraically $h$-closed in the class of topological
inverse semigroups. Also we prove that a topological semigroup $S$ with
countably compact square $S\times S$ does not contain the semigroup
$\mathscr{I}_\lambda^n$ for infinite cardinal $\lambda$ and show that the Bohr
compactification of an infinite topological symmetric inverse semigroup of
finite transformations $\mathscr{I}_\lambda^n$ of the rank $\leqslant n$ is the
trivial semigroup.
