Discretization of topological spaces

TL;DR

This paper introduces a dual notion of discretization in topology, constructing a functor from $ ext{α}$-scattered Stonean spaces to discrete spaces, which is the categorical opposite of compactification.

Contribution

It defines a new discretization concept dual to compactification and constructs a functor linking specific topological spaces to discrete spaces, expanding categorical topology.

Findings

A discretization functor from $ ext{α}$-scattered Stonean spaces to discrete spaces.

The discretization functor is the categorical dual to the Stone-ech compactification.

Interpretations of discretization at the algebraic level of functions.

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.