  The paper is a complement to the survey: M.I.Ostrovskii "To\-po\-lo\-gies on
the set of all subspaces of a Banach space and related questions of Banach
space geometry", Quaestiones Math. (to appear). It contains proofs of some
results on stability of properties of Banach spaces with respect to the
geometric opening stated in the survey without proofs.
  Some results of the present paper are of independent interest, in particular
the description of a predual property of the Banach--Saks property.
