  This article introduces the {\it strongly hit and far-miss as well as hit and
strongly far miss hypertopologies on $\textrm{CL}(X)$ associated with}
${\mathscr{B}}$, a nonempty family of subsets on the topological space $X$.
They result from the strong farness and strong nearness proximities. The main
results in this paper stem from the Hausdorffness of $(\textrm{CL}(X),
\tau_{\doublevee, \mathscr{B}})$ and $(\textrm{RCL}(X),
\tau^\doublewedge_\mathscr{B} ) $, where $\textrm{RCL}(X)$ is the space of
regular closed subsets of $X$. To obtain the results, special local families
are introduced.
