Discretization of topological spaces

Abstract

There are several compactification procedures in topology, but there is only one standard discretization, namely, replacing the original topology with the discrete topology. We give a notion of discretization which is dual (in categorical sense) to compactification and give examples of discretizations. Especially, a discretization functor from the category of $\alpha$-scattered Stonean spaces to the category of discrete spaces is constructed which is the converse of the Stone-\v{C}ech compactification functor. The interpretations of discretization in the level of algebras of functions are given.