Extentability of Automorphisms of Generic Substructures

We show that if g is a generic (in the sense of Baire category) isometry of a generic subspace of the Urysohn metric space U, then g does not extend to a full isometry of U. The same holds for the Urysohn sphere S. Let M be a Fraisse L-structure, where L is a relational countable language and M has no algebraicity. We provide necessary and sufficient conditions for the following to hold: for a generic substructure A of M, every automorphism f in Aut(A) extends to a full automorphism f in Aut(M). From our analysis, a dichotomy arises and some structural results are derived that, in particular, apply to omega-stable Fraisse structures without algebraicity.