A combinatorial property of 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.

7 Pages, submitted