On the Structure Topology on the Set of all Extreme Points of the Closed Unit Ball of the Dual of a Banach Space

Let $X$ be a real Banach space, and let $E_{X^*}$ stand for the set of all extreme points of the closed unit ball of $X^*$, endowed with the Alfsen-Effros structure topology [see E.\,M.\,Alfsen and E.\,G.\,Effros, {\it Structure in real Banach spaces I, II}, Annals of Math. 96 (1972) 98--128; ibid.\ 96 (1972) 129--73]. The fact that, for a given $s^* \in E_{X^*}$, the set $\{\pm s^* \}$ is structurally open can be characterized in many apparently different ways, whenever $X$ is nice. (We recall that $X$ is said to be nice if every extreme operator from any Banach space to $X$ is a nice operator, i.e. its adjoint preserves extreme points.) As a consequence, we obtain new characterizations (as well as new proofs of known characterizations) of those nice Banach spaces which are isometrically isomorphic to $c_0(I)$ for some set $I$.