# The strength of compactness for countable complete linear orders

## TL;DR

This paper explores the logical strength of compactness in countable complete linear orders within reverse mathematics, showing it varies with the formulation of open covers, and establishes equivalences with well-known subsystems.

## Contribution

It clarifies how the formulation of compactness affects its logical strength, providing a nuanced understanding in reverse mathematics.

## Key findings

- Compactness with uniform open covers is equivalent to WKL_0.
- Without uniformity, compactness is equivalent to ACA_0.
- Answers a question posed by François Dorais.

## Abstract

We investigate the statement "the order topology of every countable complete linear order is compact" in the framework of reverse mathematics, and we find that the statement's strength depends on the precise formulation of compactness. If we require that open covers must be uniformly expressible as unions of basic open sets, then the compactness of complete linear orders is equivalent to $\mathsf{WKL}_0$ over $\mathsf{RCA}_0$. If open covers need not be uniformly expressible as unions of basic open sets, then the compactness of complete linear orders is equivalent to $\mathsf{ACA}_0$ over $\mathsf{RCA}_0$. This answers a question of Fran\c{c}ois Dorais.