Connectedness in ditopological texture spaces
For \(S\) a set, let \({\mathcal L}\) be a completely distributive sublattice of the power set of \(S\) (with unions as suprema and with intersections as infinia) which separates points of \(S\), i.e., given \(s_1\neq s_2\) in \(S\), there exists \(L_i\in{\mathcal L}\) such that \(s_i\in L_i\) and \(s_j\not\in L_i\) for \(i,j= 1,2\), \(i\neq j\). Then \(({\mathcal L},\tau,\kappa)\) is called a ditopological texture space, if \(\tau\subset{\mathcal L}\) (resp., \(\kappa \subset{\mathcal L}\)) satisfies the three axioms of a topology (resp., of the family of closed sets). Among others, the concept of a connectedness for such spaces is introduced and some of its properties (such as productivity) are established. Relations with fuzzy topological spaces are shown.
scientific article; zbMATH DE number 1400235