Superextension as bitopological space

Supercompact space of maximal linked systems of topological space (superextension) and its subspace consisting of ultrafilters of the family of closed sets are considered. Some relations connecting above-mentioned spaces and some corollaries relating to Wallman extension in the case of $T_1$-space are obtained. For this case, some representations of sets in the space of generalized elements (defined as closed ultrafilters) for an abstract attainability problem under constraints of asymptotic character are considered. A more general variant of the above-mentioned relations for arbitrary initial topological space is also investigated (construction that uses closed ultrafilters of initial topological space is considered). Along with equipment with topology of Wallman type, topology similar to one applied for Stone compactum is used. As a result, bitopological space of maximal linked systems and corresponding bitopological space of closed ultrafilters as its subspace are realized.