Selection pointwise-maximal spaces

Let X be a Hausdorff space, and let F(X) be the set of all non-empty closed subsets of X. We say that a point p∈X is selection maximal if there exists a Vietoris continuous selection f:F(X)→X such that p∈S∈F(X) implies f(S)=p, and we say that X is a selection pointwise-maximal space if any point of X is selection maximal. The paper contains several characterizations of selection pointwise-maximal spaces which provide natural explanations about the genesis of selection maximal points.