Raney algebras and duality for \(T_0\)-spaces
Complete completely distributive lattices and related ordered structures have a rich theory. They have been extensively studied from various different perspectives. A Raney lattice is a complete completely distributive lattice that is generated by its completely join prime elements. A Raney algebra is a Raney lattice with an interior operator whose fixpoints completely generate the lattice. In this paper, the authors adapt the treatment of topological spaces via Kuratowski closure and interior operators on powersets to the setting of \(T_0\)-spaces. Their approach is to work with lattices of upsets instead of powerset algebras. It is shown that there is a dual adjunction between the category of topological spaces and the category of Raney algebras that restricts to a dual equivalence between \(T_0\)-spaces and Raney algebras. The underlying idea is to take the lattice of upsets of the specialization order with the restriction of the interior operator of a space as the Raney algebra associated to a topological space. Some further properties of topological spaces are explored in the dual setting of Raney algebras. In particular, spaces that are \(T_1\) correspond to Raney algebras whose underlying lattices are Boolean, and Alexandroff \(T_0\)-spaces correspond to Raney algebras whose interior operator is the identity. Algebraic description of sober spaces results in algebraic considerations that lead to a new type of weakly sober spaces -- almost sober spaces lying strictly between \(T_0\) spaces and sober spaces. Such spaces may deserve further investigation.
scientific article; zbMATH DE number 7347071