Raney Algebras and Duality for $$T_0$$-Spaces

Abstract

In this note we adapt the treatment of topological spaces via Kuratowski closure and interior operators on powersets to the setting of $$T_0$$ -spaces. 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. 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. Further properties of topological spaces are explored in the dual setting of Raney algebras. 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 generalization of sober that lies strictly between $$T_0$$ and sober.