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.