  We propose a new class of hypertopologies, called here weak$^{\ast }$
hypertopologies, on the dual space $\mathcal{X}^{\ast }$ of a real or complex
topological vector space $\mathcal{X}$. The most well-studied and well-known
hypertopology is the one associated with the Hausdorff metric for closed sets
in a complete metric space. Therefore, we study in detail its corresponding
weak$^{\ast }$ hypertopology, constructed from the Hausdorff distance on the
field (i.e. $\mathbb{R}$ or $\mathbb{C}$) of the vector space $\mathcal{X}$ and
named here the weak$^{\ast }$-Hausdorff hypertopology. It has not been
considered so far and we show that it can have very interesting mathematical
connections with other mathematical fields, in particular with mathematical
logics. We explicitly demonstrate that weak$^{\ast }$ hypertopologies are very
useful and natural structures\ by using again the weak$^{\ast }$-Hausdorff
hypertopology in order to study generic convex weak$^{\ast }$-compact sets in
great generality. We show that convex weak$^{\ast }$-compact sets have
generically weak$^{\ast }$-dense set of extreme points in infinite dimensions.
An extension of the well-known Straszewicz theorem to Gateaux-differentiability
(non necessarily Banach) spaces is also proven in the scope of this
application.
