Computational approximations of compact metric spaces

TL;DR

This paper introduces a method to approximate compact metric spaces using inverse sequences of finite topological spaces, enabling the computation of algebraic invariants like homology groups, with practical implementation considerations.

Contribution

It presents a novel approach to approximate compact metric spaces via inverse sequences, facilitating the computation of their algebraic invariants.

Findings

Inverse sequences approximate compact metric spaces effectively.

Homology groups can be computed from these approximations.

The method's computational implementation is feasible.

Abstract

Given a compact metric space $X$, we associate to it an inverse sequence of finite $T_0$ topological spaces. The inverse limit of this inverse sequence contains a homeomorphic copy of $X$ that is a strong deformation retract. We provide a method to approximate the homology groups of $X$ and other algebraic invariants. Finally, we study computational aspects and the implementation of this method.