Classes of Banach spaces stable and unstable with respect to the opening

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.