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Partition relations for countable topological spaces. (English) Zbl 0647.05005

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.

MSC:

05A17 Combinatorial aspects of partitions of integers
54F05 Linearly ordered topological spaces, generalized ordered spaces, and partially ordered spaces
03E05 Other combinatorial set theory
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References:

[1] Devlin, D., Some Partition Theorems and Ultrafilters on ω, (Ph.D. dissertation (1979), Dartmouth College)
[2] Erdös, P.; Hajnal, A., Unsolved and solved problems in set theory, (Proc. Sympos. Pure Math., Vol. 25 (1974), Amer. Math. Soc: Amer. Math. Soc Providence, R. I), 267-287
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