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This article introduces the {\it strongly hit and far-miss as well as hit andstrongly 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 mainresults 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 ofregular closed subsets of $X$. To obtain the results, special local familiesare introduced.

Strongly Hit and Far Miss Hypertopology and Hit and Strongly Far Miss Hypertopology