On closedness assumptions in selection theorems

We begin by a short survey of various attempts in selection theory to avoid the closedness assumption for values of multivalued mappings. We collect special cases when Michael's Gδ-problem admits an affirmative solution and we prove some unified theorems of such type. We also show that in general this problem has a negative solution. In comparison with a recent result of Filippov, we work directly in the Hilbert cube rather than in the space of all probabilistic measures endowed with different topologies.