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
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