A combinatorial property of rho-functions

TL;DR

This paper proves that for any Hausdorff topology on the first uncountable ordinal, subsets homeomorphic to the rationals can be refined to preserve a shift-increasing property of a specific rho-function.

Contribution

It introduces a method to refine rational-like subsets in uncountable topologies to maintain a shift-increasing rho-function property.

Findings

Any subset homeomorphic to the rationals can be refined to a shift-increasing rho-function.

The result applies to all Hausdorff topologies on ω₁.

Provides a new combinatorial property related to rho-functions.

Abstract

We show that if $\mathcal{T}$ is any Hausdorff topology on ${\omega}_{1}$, then any subset of ${\omega}_{1}$ which is homeomorphic to the rationals under $\mathcal{T}$ can be refined to a homeomorphic copy of the rationals on which $\bar{\rho}$ is shift-increasing.