Total orders on compact metric spaces and covering dimension

TL;DR

This paper establishes an equivalence between the finite covering dimension of a compact metric space and the existence of a total order with finite snake number, linking topological dimension to order-theoretic properties.

Contribution

It introduces a novel characterization of finite covering dimension via total orders with finite snake number in compact metric spaces.

Findings

Finite covering dimension is equivalent to the existence of a total order with finite snake number.

Provides a new order-theoretic perspective on topological dimension.

Bridges concepts between topology and order theory.

Abstract

We prove that for a compact metric space the property of having finite covering dimension is equivalent to the existence of a total order with finite snake number.