On the lattice of Fréchet-Nikodým topologies (Q1820886)

Let \({\mathcal A}\) be an algebra of subsets of a set X. It is known that the set FN(\({\mathcal A})\) of all FN-topologies on \({\mathcal A}\) is a distributive complete lattice, which is not complemented for \({\mathcal A}\) being infinite. The author proves that, for a topology \(G\in FN({\mathcal A})\), there is a set \(A\in {\mathcal A}\) such that the restriction \(G| {\mathcal A}\cap A\) of G on \({\mathcal A}\cap A\) is the discrete topology and \(G| {\mathcal A}\cap (X\setminus A)\) is the indiscrete topology iff G has a complement in \(FN({\mathcal A})\). In the following part the author studies pseudocomplements in \(FN({\mathcal A})\); the pseudocomplement \(G^ c\) of \(G\in FN({\mathcal A})\) is defined as \(G^ c=\sup \{H\in FN({\mathcal A}):\inf \{G,H\}=0\}\).