Partition relations for countable topological spaces (Q1104325)

We consider partition relations for pairs of elements of a countable topological space. For spaces with infinitely many nonempty derivatives a strong negative theorem is obtained. For example, it is possible to partition the pairs of rationals into countably many pieces so that every homeomorph of the rationals contains a pair from every piece. Some positive results are also proved for ordinal spaces of the form \(\omega^{\alpha}+1\), where \(\alpha\) is countable.