Definability of Geometric Properties in Algebraically Closed Fields.

We prove that there exists no sentence F of the language of rings with an extra binary predicat I(2) satisfying the following property: for every definable set X subset of or equal to C(2), X is connected if and only if (C, X) satisfies F, where It is interpreted by X. We conjecture that the same result holds for closed subset of C(2). We prove some results motivated by this conjecture.