Symmetric inverse topological semigroups of finite rank ≤ n

We establish topological properties of the symmetric inverse topological semigroup of finite transformations of the rank ≤ n . We show that the topological inverse semigroup is algebraically h -closed in the class of topological inverse semigroups. Also we prove that a topological semigroup S with countably compact square S × S does not contain the semigroup for infinite cardinal λ and show that the Bohr compactification of an infinite topological symmetric inverse semigroup of finite transformations of the rank ≤ n is the trivial semigroup.