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

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 $(\textrm{CL}(X), τ_{\doublevee, \mathscr{B}})$ and $(\textrm{RCL}(X), τ^\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.

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