Modifications of product convergence structures (Q2785267)

The author considers a convergence structure determined on a set \(X\) as a function \(q\) from the set \(F(X)\) of all filters on \(X\) into the set \(P(X)\) of all subsets of \(X\) satisfying some minimal conditions. The sequential strengthenings of these conditions allow to define pseudotopological, pretopological and topological convergence structures, to receive the pseudotopological modification \(\rho (q)\), the pretopological modification \(\pi (q)\), and the topological modification \(\lambda (q)\) of \(q\). It shows that \(\rho (q)\) is the finest pseudotopology coarser than \(q\), \(\pi (q)\) is the finest pretopology coarser than \(q\), \(\lambda (q)\) is the finest topology coarser than \(q\), \(\lambda (q)\leq\pi (q)\leq \rho (q)\leq q\). The author proves multiplicativity of topological and pseudotopological convergence structures, multiplicativity of pseudotopological modifications and some relations between different products.