Abstract

We show how we can characterize T0 -topological spaces in terms of preorders describing a base for the space. In particular, we show how any T0-topological space  can be represented as the space whose points are the neighborhood filters of one of its basis for the open sets. Conversely, we prove that  any dense family of filters on a preorder defines a topological space whose characteristics are strictly connected to the ones of the preorder. Therefore, we analyze how the separation properties of the topological space can be described in terms of the algebraic properties of the corresponding preorder and family of filters.

Furthermore, we outline the algebraic conditions on a selected base of the topological space ensuring that the space is compact and T1. This allows us to establish a duality between the category of compact T1 spaces with continuous closed maps and an appropriate category of lattices.

Moreover, we could specialize this duality to the category of compact Hausdorff spaces with continuous maps. 

These characterizations allow us to give a description of the Stone-Čech compactification of a topological space in terms of lattices.