Given a cardinal $kappa$ and a sequence $left(alpha_iright)_{iinkappa}$ of ordinals, we determine the least ordinal $beta$ (when one exists) such that the topological partition relation [betarightarrowleft(top,alpha_iright)^1_{iinkappa}] holds, inclu
ding an independence result for one class of cases. Here the prefix $top$ means that the homogeneous set must have the correct topology rather than the correct order type. The answer is linked to the non-topological pigeonhole principle of Milner and Rado.
